Delegation in Veto Bargaininglowast

Navin Kartikdagger Andreas KleinerDagger Richard Van Weeldensect

June 11 2020

Abstract

A proposer requires the approval of a veto player to change a status quo Preferences

are single peaked Proposer is uncertain about Vetoerrsquos ideal point We study Proposerrsquos

optimal mechanism without transfers Vetoer is given a menu or a delegation set to

choose from The optimal delegation set balances the extent of Proposerrsquos compromise

with the risk of a veto Under reasonable conditions ldquofull delegationrdquo is optimal Vetoer

can choose any action between the status quo and Proposerrsquos ideal action This outcome

largely nullifies Proposerrsquos bargaining power Vetoer frequently obtains her ideal point

and there is Pareto efficiency despite asymmetric information More generally we identify

when ldquointerval delegationrdquo is optimal Optimal interval delegation can be a Pareto im-

provement over cheap talk We derive comparative statics Vetoer receives less discretion

when preferences are more likely to be aligned by contrast to expertise-based delegation

Methodologically our analysis handles stochastic mechanisms

lowastWe thank Nageeb Ali Wiola Dziuda Alex Frankel Sanjeev Goyal Marina Halac Elliot Lipnowski MalleshPai Mike Ting and various seminar and conference audiences for helpful comments Bruno Furtado providedexcellent research assistance

daggerDepartment of Economics Columbia University Email nkartikcolumbiaeduDaggerDepartment of Economics Arizona State University Email andreaskleinerasuedusectDepartment of Economics University of Pittsburgh Email rmv22pittedu

1 Introduction

Motivation There are numerous situations in which one agent or group can make propos-als but another must approve them Legislatures (eg US Congress) send bills to executives(eg the President) who can veto them Governmental legislation can be struck down as un-constitutional by the judiciary Prosecutors choose which charges to bring against defendantsbut judges and juries decide whether to convict A real-estate agent can recommend a houseto his client but the client must decide to put in an offer similarly a search committee canput forward a candidate but the organization decides whether to hire her

Romer and Rosenthal (1978) present a seminal analysis of such veto bargaining Theirframework is one of complete information in which a proposer (Congress government pros-ecutor salesperson search committee) makes a take-it-or-leave-it proposal to a veto player(President judiciary judgejury customer organization) Preferences are single peaked Thatis rather than ldquodividing a dollarrdquo the negotiating parties share some preference alignmentSuch preferences are plausible in the contexts mentioned above

Our paper studies veto bargaining with incomplete information The proposer is uncer-tain about the veto playerrsquos preferencesmdashspecifically which proposals would actually getvetoed Previous scholars have emphasized this featurersquos importance see Cameron and Mc-Cartyrsquos (2004) survey But to our knowledge our paper is the first that takes a general ap-proach to the issue We do not assume the proposer is restricted to making a single proposal(eg Romer and Rosenthal 1979) nor do we fix any particular negotiating protocol (eg oneround of cheap talk in Matthews 1989 Forges and Renault 2020) Instead we consider thepossible outcomes across all possible protocols by taking a mechanism design approach Ourfocus is on identifying the proposerrsquos optimum

There are at least two reasons this mechanism design approach is of interest First it iden-tifies an upper bound on the proposerrsquos welfare Second as is standard in settings withouttransfers any (deterministic) mechanism is readily interpreted and implemented as delega-tion That is the proposer simply offers a menu of options the veto player can select anyone or reject them all Menus or delegation sets are observed in practice in some applicationsof our model Salespeople show customers subsets of products and search committees putforward multiple candidates for their organization to choose among In politics a bill autho-rizing at most $x of spending effectively offers an executive who controls the implementingbureaucracy the choice of any spending level in the interval [0 x] Bills can also grant more orless discretion of how to allocate a given level of spending as encapsulated by former SenatorRuss Feingold in the context of the recent US coronavirus stimulus package ldquoCongress has

1

to decide how much discretion it wants to delegate to executive branch officialsrdquo (Washing-ton Post March 22 2020)

Main results We formally study a one-dimensional environment There is status quo policyor action 0 that obtains if there is a veto There are two agents Proposerrsquos ideal action is 1Vetoerrsquos ideal action v is her private information We assume Vetoer preferences are repre-sented by a quadratic loss function but allow Proposer to have any concave utility function1

Our first result (Proposition 1) identifies conditions under which it is optimal for Proposerto simply let Vetoer choose her preferred action in the interval [0 1] We call this full delegationbecause Proposer only excludes options that are from his point of view dominated by simplyoffering his ideal action 1 no matter Vetoerrsquos ideal point Intuitively full delegation is optimalwhen the specter of a veto looms large in particular it is sufficient that the density of Vetoerrsquosideal point is decreasing on the unit interval Optimality of full delegation is quite strikingdespite Proposer having considerable bargaining and commitment power it is Vetoer whofrequently gets her first best This is a telling manifestation of private informationrsquos conse-quences Unlike in most other settings that confer information rents however full delegationimplies ex-post Pareto efficiency

The opposite of full delegation is no compromise Proposer only offers his ideal action 1Of course Vetoer can veto and choose the status quo 0 Proposition 2 gives conditions foroptimality of no compromise It is sufficient for example that Proposer has a linear lossfunctionmdashso in the relevant region of actions [0 1] he only cares about the mean actionmdashand the density of Vetoerrsquos ideal point is increasing in this region This case juxtaposes nicelyagainst the aforementioned decreasing-density condition for full delegation

Both full delegation and no compromise are boundary cases of interval delegation Proposeroffers a menu of the form [c 1] Interval delegation is interesting for multiple reasons Amongthem are that such delegation sets are simple to interpret and implement and they turn out tobe tractable for comparative statics Proposition 3 provides conditions under which intervaldelegation is optimal These are met in particular when Proposer has a linear or quadraticloss function and Vetoerrsquos ideal point distribution is logconcave (Corollary 3)

We show that under reasonable conditions optimal interval delegation yields a Paretoimprovement over singleton proposals even when cheap talk is allowed and full delegation

1 Proposerrsquos risk attitude is important because he faces uncertainty about the final action Among determin-istic mechanisms Vetoerrsquos risk attitude is irrelevant so quadratic loss entails no restriction beyond symmetryaround the ideal point

2

to decide how much discretion it wants to delegate to executive branch officialsrdquo (Washing-ton Post March 22 2020)

Main results We formally study a one-dimensional environment There is status quo policyor action 0 that obtains if there is a veto There are two agents Proposerrsquos ideal action is 1Vetoerrsquos ideal action v is her private information We assume Vetoer preferences are repre-sented by a quadratic loss function but allow Proposer to have any concave utility function1

Our first result (Proposition 1) identifies conditions under which it is optimal for Proposerto simply let Vetoer choose her preferred action in the interval [0 1] We call this full delegationbecause Proposer only excludes options that are from his point of view dominated by simplyoffering his ideal action 1 no matter Vetoerrsquos ideal point Intuitively full delegation is optimalwhen the specter of a veto looms large in particular it is sufficient that the density of Vetoerrsquosideal point is decreasing on the unit interval Optimality of full delegation is quite strikingdespite Proposer having considerable bargaining and commitment power it is Vetoer whofrequently gets her first best This is a telling manifestation of private informationrsquos conse-quences Unlike in most other settings that confer information rents however full delegationimplies ex-post Pareto efficiency

The opposite of full delegation is no compromise Proposer only offers his ideal action 1Of course Vetoer can veto and choose the status quo 0 Proposition 2 gives conditions foroptimality of no compromise It is sufficient for example that Proposer has a linear lossfunctionmdashso in the relevant region of actions [0 1] he only cares about the mean actionmdashand the density of Vetoerrsquos ideal point is increasing in this region This case juxtaposes nicelyagainst the aforementioned decreasing-density condition for full delegation

Both full delegation and no compromise are boundary cases of interval delegation Proposeroffers a menu of the form [c 1] Interval delegation is interesting for multiple reasons Amongthem are that such delegation sets are simple to interpret and implement and they turn out tobe tractable for comparative statics Proposition 3 provides conditions under which intervaldelegation is optimal These are met in particular when Proposer has a linear or quadraticloss function and Vetoerrsquos ideal point distribution is logconcave (Corollary 3)

We show that under reasonable conditions optimal interval delegation yields a Paretoimprovement over singleton proposals even when cheap talk is allowed and full delegation

1 Proposerrsquos risk attitude is important because he faces uncertainty about the final action Among determin-istic mechanisms Vetoerrsquos risk attitude is irrelevant so quadratic loss entails no restriction beyond symmetryaround the ideal point

2

is not optimal (Proposition 5) We trace the intuition to Proposer being more willing to com-promise when he can offer Vetoer an interval of options rather than only singletons

We develop two comparative statics restricting attention to interval delegation mdash eitherjustified by optimality or otherwise First what happens when Proposer becomes more riskaverse Proposition 4(i) establishes that Vetoer is given more discretion the optimal thresholdin the interval delegation set (ie that denoted c above) decreases Intuitively Proposer offersa larger set of options to mitigate the risk of a veto Second what about when Vetoer becomesmore ex-ante aligned with Proposer Formally we consider right shifts in Vetoerrsquos ideal pointdistribution in the sense of likelihood ratio dominance Proposition 4(ii) establishes that dis-cretion decreases the optimal interval delegation threshold increases Intuitively this is be-cause Proposer is less concerned that a veto will occur Although these comparative staticsappear natural it is the structure of interval delegation that allows us to establish them

Contrast with expertise-based delegation The second comparative static mentioned abovecontrasts with a key theme of the expertise-based delegation literature following Holmstrom(1977 1984) In that literature an agent is given discretion over actions because her private in-formation is valuable to the principal the principal limits the degree of discretion because ofpreference misalignment One version of the so-called Ally Principle says that a more alignedagent receives more discretion Holmstrom (1984) establishes its validity under reasonablygeneral conditions so long as delegation sets take the form of intervals2 In our setting thereason Proposer gives Vetoer discretion is fundamentally different from that in Holmstromit is not to benefit from Vetoerrsquos expertise rather Proposer trades off the risk of a veto withthe extent of compromise (In jargon our delegator has state-independent preferences bycontrast to the state-dependent preferences in most of the literature following Holmstrom)Hence we find less discretion emerging when there is in a suitable sense more ex-ante pref-erence alignment

Methodology We hope some readers will find our analysis interesting on a methodologicallevel While it is convenient and economically insightful to describe our substantive resultsin terms of optimal delegation sets the formal problem we study is one of mechanism de-sign without transfers Our analytical methodology builds on the infinite-dimensional Lan-grangian approach advanced by Amador Werning and Angeletos (2006) and Amador andBagwell (2013) Unlike these authors and many others including the important contributions

2 The comparative static may fail absent interval delegation (eg Alonso and Matouschek 2008)

3

by Melumad and Shibano (1991) and Alonso and Matouschek (2008) we also cover stochasticmechanisms3 That is we allow for mechanisms in which Vetoer may choose among lotteriesover actions We view such mechanisms as not only theoretically important but also relevantin applications For instance in the hiring application mentioned earlier the search committee(Proposer) can offer the organization (Vetoer) an option of hiring ldquothe best available candidatewith at least five yearsrsquo work experience in that countryrdquo Both parties would view this optionas a lottery over some subset of candidates

Stochastic mechanisms can sometimes be optimal in our framework Nevertheless weestablish that our sufficient conditions for full delegation no compromise and interval dele-gation (Propositions 1 2 and 3) ensure optimality of these (deterministic) mechanisms evenamong stochastic mechanisms Furthermore by permitting stochastic mechanisms our suffi-cient conditions are shown to also be necessary for a class of Proposerrsquos utility functions thatinclude linear and quadratic loss Our approach to handling stochastic mechanisms shouldbe useful in other delegation problems

Recently Kolotilin and Zapechelnyuk (2019) have introduced balanced delegation problemswhich are delegation problems in which certain extreme actions or outside options must beincluded Our setting fits into their general framework as one can assume the status quomust be part of the delegation set Kolotilin and Zapechelnyuk derive a general equivalencebetween such problems and monotone Bayesian persuasion problems More concretely theyshow how some results from the latter literature (eg Kolotilin 2018 Dworczak and Martini2019) can be brought to bear on ldquolinearrdquo balanced delegation problems4 Our approach ofdirectly studying the delegation problem is complementary and has some advantages Firstit permits insights absent said linearity this is most evident in our full delegation resultSecond we believe it provides some more transparent economic intuitions Third unlikeKolotilin and Zapechelnyuk (2019) we can address stochastic mechanisms and necessity ofour sufficient conditions At a broader level note that by contrast to us Kolotilin and Za-pechelnyuk (2019) highlight applications concerning expertise-based delegation (ie withstate-dependent delegator preferences) Zapechelnyuk (2019) applies their methodology toa quality certification problem that he shows maps into a delegation problem in which the

3 A qualification is appropriate both Amador et al (2006) and Amador and Bagwell (2013) allow for moneyburning identifying conditions under which optimal mechanisms do not employ that instrument see AmadorBagwell and Frankel (2018) as well Stochastic mechanisms are equivalent to money burning for certain prefer-ence specifications but in general they are not equivalent Ambrus and Egorov (2017) discuss settings in whichmoney burning can be optimal

4 This linearity requires that the utilities of Proposer and (all types of) Vetoer viewed as a function of theaction have the same curvature

4

delegatorrsquos preferences are state independent

Like us Amador and Bagwell (2019) directly analyze a delegation problem with an out-side option They focus on monopoly regulation (Baron and Myerson 1982) without trans-fers Unlike in our paper their delegatorrsquos preferences are state dependent and they do notconsider stochastic mechanisms or necessity

Outline The rest of the paper proceeds as follows Section 2 presents our model Section 3contains our main results on the conditions for optimality of full delegation no compromiseand more broadly interval delegation Section 4 develops comparative statics and makescomparisons with other mechanisms Section 5 discusses some applications Section 6 con-cludes All proofs are in the appendices

2 Model

We consider a classic bargaining problem between two players a proposer (he) and a vetoplayer (she) who jointly determine a policy outcome or action a isin R In a manner elaboratedbelow Proposer makes a proposal that Vetoer can either accept or reject If Vetoer rejects astatus-quo action is preserved we normalize the status quo to 0

We assume both players have single-peaked utilities Proposerrsquos utility is u(a) that is con-cave maximized uniquely at a = 1 (essentially a normalization) and twice differentiable atall a ∕= 15 Unless indicated explicitly we use lsquoconcaversquo lsquoincreasingrsquo etc to mean lsquoweaklyconcaversquo lsquoweakly increasingrsquo etc We will sometimes invoke a restriction to the followingsubclass of Proposer preferences which stipulates a convex combination of the widely-usedlinear and quadratic loss functions

Condition LQ For some γ isin [0 1]

u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Vetoerrsquos utility is represented by minusl(|v minus a|) where l(middot) is strictly increasing So her utilityis symmetric around the unique ideal point v For tractability we assume l(|vminus a|) = (vminus a)2A subset of our results will rely only on Vetoerrsquos ordinal preferences for which the choiceof quadratic loss entails no loss of generality given that Vetoerrsquos utility is symmetric around

5 Permitting a point of nondifferentiability allows the linear loss function u(a) = minus|1 minus a| When we writeuprime(1) subsequently it refers to the left-derivative when u is not differentiable at 1

5

her ideal point Specifically Vetoerrsquos ordinal preferences are sufficient when we consider onlydeterministic mechanisms

A key ingredient of our model is that v is Vetoerrsquos private information We accordinglyrefer to v as Vetoerrsquos type It is drawn from a cumulative distribution F whose support is aninterval [v v] where we permit v = minusinfin andor v = infin We assume F admits a continuouslydifferentiable density f and that f(middot) gt 0 on [0 1] All aspects of the environment except thetype v are common knowledge If v were common knowledge this model would reduce tothat of Romer and Rosenthal (1978)

Naturally it is in Proposerrsquos interests to elicit information from Vetoer about v For exam-ple they might engage in cheap talk communication (Matthews 1989) possibly over multiplerounds or Proposer might make sequential proposals and so on To circumvent issues aboutexactly how the bargaining ensues we take a mechanism design approach Following the rev-elation principle we consider direct revelation mechanisms hereafter simply mechanisms

A deterministic mechanism is described by a real-valued function α(v) which specifies theaction when Vetoerrsquos type is v and must satisfy the usual incentive compatibility (IC) andindividual rationality (IR) conditions IC requires that each type v prefers α(v) to α(vprime) for anyvprime ∕= v IR requires that each type v prefers α(v) to the status quo 0 Notice that any determin-istic mechanism is equivalent to the Proposer offering a (closed) menu or delegation set A sube Rand Vetoer choosing an action from A cup 0 We will also consider the more general classof stochastic mechanisms which specify probability distributions over actions for each Vetoertype with analogous IC and IR constraints to those aforementioned Stochastic mechanismsare theoretically important because the revelation principle does not justify focussing only ondeterministic mechanisms As noted in the introduction they may also be relevant for appli-cations A notable contribution of this paper is to establish conditions under which despitethe absence of transfers stochastic mechanisms cannot improve upon deterministic ones6

We highlight that our model is one of private values Vetoerrsquos type does not directly affectProposerrsquos preferences This is by way of contrast with the delegation literature initiated byHolmstrom (1984) in which a principal gives discretion to an agent because of the agentrsquosexpertise ie because they have interdependent preferences We could extend our modeland analysis to incorporate this expertise-based delegation or discretion aspect but one ofour main themes is that discretion will emerge even when that is absent

6 Remark 2 below explains why stochastic mechanisms can be optimal Example E1 in Appendix E elaboratesAlonso and Matouschek (2008 p 281) provide a related example in their framework without a veto option seealso Kovac and Mylovanov (2009 Section 4)

6

As mentioned in the introduction the mechanism design approach we take can be viewedas identifying an upper bound on Proposerrsquos welfare That said as also mentioned there wefind the implementation via delegation sets quite realistic in various contexts

21 Proposerrsquos Problem

We now formally define Proposerrsquos problem Let M(R) denote the set of Borel probabil-ity distributions on R7 and M0(R) be the subset of distributions with finite expectation andfinite variance Denote by δa the degenerate distribution that puts probability 1 on action a Astochastic mechanismmdashor simply a mechanism without qualificationmdashis a measurable func-tion m [v v] rarr M0(R) with m(v) being the probability distribution over actions for typev8 To reduce notation for any deterministic mechanism α [v v] rarr R we also denote themechanism v 983041rarr δα(v) by α For any integrable function g A rarr R let Em(v)[g(a)] denote theexpectation of g(a) when a has distribution m(v) We only consider mechanisms m for whichv 983041rarr Em(v)[a] is integrable Define the subset of mechanisms

S =983051m [v v] rarr M0(R) | m(0) = δ0 and forallv lt vprime Em(v)[a] le Em(vprime)[a]

983052

That is S consists of mechanisms in which type 0 gets the status quo and a higher type re-ceives a higher expected action The first requirement is implied by IR since Vetoer can al-ways choose the status quo The second is implied by IC since Vetoerrsquos utility minus(v minus a)2 isequivalently represented by av minus a22 singlecrossing difference in (a v) yields monotonicityof Em(v)[a] in v from standard arguments (elaborated in fn 9 below)

Proposerrsquos problem is

maxmisinS

983133Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046=

983133 v

0

Em(s)[a]ds forallv isin [v v] (IC-env)

As noted above it is without loss to restrict attention to mechanisms in S The constraint(IC-env) captures the additional content of IC beyond monotonicity via an analog of thestandard envelope formula9 Note that since IC requires that no type prefer type 0rsquos lottery

7 We endow M(R) with the topology of weak convergence and the corresponding Borel σ-algebra8 There is no loss in restricting attention to M0(R) instead of M(R) because no type would choose a lottery

with infinite mean or variance given that the status quo is available9 Formally using quadratic utility IC requires forallv vprime Em(v)[avminusa22] ge Em(vprime)[avminusa22] and IR requires forallv

Em(v)[avminus a22] ge 0 An IC mechanism m thus satisfies IR if and only if m(0) = δ0 It follows that m satisfies IC

7

over its own and type 0rsquos IR constraint requires that it receive action 0 (captured in S) everytypersquos IR constraint is implied by type 0rsquos An optimal mechanism is a solution to problem (P)

If we restricted attention to deterministic mechanisms the analogous problem for Pro-poser would be

maxαisinA

983133u(α(v)) dF (v) (D)

st vα(v)minus α(v)22 =

983133 v

0

α(s)ds

whereA = α [v v] rarr R | α(0) = 0 and α is increasing

Any deterministic mechanism α that is IC has a corresponding (closed) delegation setAα =

983126v α(v) Conversely any delegation set A has a corresponding deterministic mecha-

nism αA where αA(v) is the action in A cup 0 that type v prefers the most (with ties brokenin favor of Proposer) Note that αA satisfies IC and IR While our formal analysis works withmechanisms it is easier and more economically intuitive to describe our main results whichconcern certain deterministic mechanisms using delegation sets

We emphasize some terminology an optimal deterministic mechanism (or an optimal dele-gation set) is a solution to problem (D) But when we say that a deterministic mechanism (ordelegation set) is optimal we mean that it solves problem (P) ie no stochastic mechanism canstrictly improve on it

Remark 1 We capture veto power via a standard interim IR constraint An alternative as inCompte and Jehiel (2009) would be to allow Vetoer to exercise her veto even after the mech-anism determines an action This is stronger than just ex-post IR because it also strengthensthe IC constraint when type v mimics type vprime v may veto a different set of allocations than vprime

would and so the action distribution that v evaluates the deviation with is not m(vprime) Whichform of veto power is conceptually appropriate depends on the application But any IC andIR deterministic mechanism also satisfies the ex-post veto constraint Hence the sufficientconditions we provide below for optimality of delegation sets would remain sufficient ⋄

and IR if and only if m isin S and the envelope formula (IC-env) holds To confirm this let V(v) = Em(v)[avminusa22]Mechanism m is IC if and only if V(v) = maxvprime Em(vprime)[av minus a22] which holds if and only if V is convex andV(v) = V(0) +

983125 v

0Em(s)[a]ds (Milgrom and Segal 2002) Consequently m is IC and IR if and only if Em(v)[a] is

increasing in v (IC-env) holds and m(0) = δ0

8

22 Preliminary Observations

Consider delegation sets Notice first that there is no loss for Proposer in including hisideal action 1 in the delegation set for any Vetoer type v either it does not affect the chosenaction or it results in a preferable action Next there is no loss for Proposer in excludingactions outside [0 1] shrinking a delegation set A that contains 1 to A cap [0 1] only results ineach type choosing an action closer to 1 We have

Lemma 1 There is an optimal delegation set A satisfying 1 isin A sube [0 1]

It would also be without loss to assume that a delegation set contains the status quo 0 Wedonrsquot do so however because it is convenient to sometimes describe optimal delegation setswithout including 0

For any a isin (0 1) the delegation set A = [a 1] strictly dominates the singleton a becauseA results in preferable actions for Proposer when v gt a This simple observation highlightsthe significance of giving Vetoer discretion despite our model shutting down the expertise-based rationale that the literature initiated by Holmstrom (1984) has focused on

While Proposer always wants to include action 1 in the delegation set he faces a tradeoffwhen including any action a isin (0 1) Allowing Vetoer to choose such an action a reduces theprobability of a veto (or any action less than a) but also reduces the probability that Vetoerchooses an action even higher than a which Proposer would prefer to a

3 Delegation and Optimal Mechanisms

31 Full Delegation

In light of Lemma 1 we refer to the delegation set [0 1] as full delegation Note that full del-egation does impose some constraints on Vetoer But the constraints are minimal only actionsoutside the convex hull of the status quo and Proposerrsquos ideal point are excluded Given theveto-bargaining institution an outcome of full delegation starkly captures how Vetoerrsquos pri-vate information can corrode Proposerrsquos bargaining or agenda-setting power Indeed if thetype distributionrsquos support is [0 1] then Vetoer gets her first best Regardless of the supporthowever full delegation results in ex-post Pareto efficiency unlike in most other settings thatconfer information rents Full delegation thus contrasts sharply with the outcome under com-plete information (Romer and Rosenthal 1978) in which case Proposer would make Vetoerwith ideal point v isin (0 12] indifferent with exercising the veto while getting his own ideal

9

action 1 from types v isin (12 1) It also contrasts with the outcome under incomplete informa-tion were Proposer restricted to making a singleton proposal In that case the proposal wouldlead to a veto by some subinterval of types v isin [0 1] hence to ex-post Pareto inefficiency andall Vetoer types would be weakly worse off many strictly10

It is thus of interest to know when full delegation is optimal The following quantityconcerning the concavity of Proposerrsquos utility will play a key role in our analysis

κ = infaisin[01)

minusuprimeprime(a)

Proposition 1 (Full delegation) Full delegation is optimal if

κF (v)minus uprime(v)f(v) is increasing on [0 1] (1)

Conversely under Condition LQ full delegation is optimal only if (1) holds

Consider sufficiency in Proposition 1 If we had restricted attention to deterministic mech-anisms and assumed that Proposerrsquos utility is a quadratic loss function then the result wouldfollow from a result in Alonso and Matouschek (2008) even though their model does nothave a veto constraint and as such highlights expertise-based delegation To make the con-nection we observe that when u(a) = minus(1 minus a)2 condition (1) is equivalent to Alonso andMatouschekrsquos ldquobackward biasrdquo being convex on [0 1] Their Proposition 2 then implies thatif 0 1 is contained in the optimal delegation set then the interval [0 1] is contained in theoptimal delegation set But recall from Lemma 1 that in choosing among delegation sets Pro-poser need not offer any action outside [0 1] and can offer his ideal point 1 moreover he mayas well offer the status quo 0 It follows that full delegation is an optimal delegation set

We therefore emphasize that Proposition 1 establishes optimality among more generalProposer preferences and stochastic mechanisms11 The proposition directly implies

Corollary 1 Full delegation is optimal if the type density is decreasing on [0 1]

In particular it is sufficient that the type density is unimodal with a negative mode Toobtain intuition for the corollary consider removing any interval (a a) from a delegation set

10 Note that vetoes can have positive probability even under full delegation this is the case if and only ifProb(v lt 0) gt 0

11 In a model without an outside option Kovac and Mylovanov (2009) provide sufficient conditions for cer-tain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic lossfunction

10

A that contains [a a] This change induces Vetoer with type v isin (a a) to choose between a anda Due to her symmetric utility function Vetoer will choose a when v isin

983043a a+a

2

983044 which harms

Proposer while Vetoer will choose a when v isin983043a+a2 a983044 which benefits Proposer When the

type density is decreasing on [a a] the former possibility is more likely in fact the pruneddelegation set induces an action distribution that is second-order stochastically dominated12

Since Proposerrsquos utility is concave he prefers the original delegation set A As any (closed)delegation set can be obtained by successively removing open intervals full delegation is anoptimal delegation set While this explanation applies among delegation sets Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms

Removing an interval increases the expected action when the type density is increasingbut it also increases the probability of a lower action Thus when Proposer is risk averseit can be optimal to not remove an interval even if the density is increasing on that intervalThis explains why condition (1) is weaker than f decreasing on [0 1] In general removing aninterval is optimal only if the density is increasing quickly relative to Proposerrsquos risk aversionThis suggests that full delegation is optimal whenever Proposer is sufficiently risk averseProposition 1 allows us to formalize the point using the Arrow-Prat (Arrow 1965 Pratt 1964)coefficient of absolute risk aversion

Corollary 2 Full delegation is optimal if Proposer is sufficiently risk averse ie if infaisin[01)

minusuprimeprime(a)uprime(a)

is sufficiently large

Consider now necessity in Proposition 1 If Proposer has a linear loss utility then our pre-ceding discussion explains why a delegation set A containing [a a] sube [0 1] should be prunedto A (a a) if the type density is increasing on this interval the expected action increasesHence the converse of Corollary 1 holds for linear loss utility For quadratic loss utility (andwith additional smoothness assumptions) Alonso and Matouschekrsquos (2008) Proposition 2 im-plies that that a weaker condition F (v)minus (1minus v)f(v) increasing on [0 1] is necessary for full

12 Let GX denote the cumulative distribution of the action induced by A GY denote that induced by A (a a)and let amid = (a+ a)2 Since F is the distribution of v it holds that GX(a) = GY (a) for a isin (a a] GX(a) =F (a) on [a a] and GY (a) = F (amid) for a isin [a a) Consequently for any a isin [a amid] GY (a) ge GX(a) and983125 a

0[GY (t)minusGX(t)] dt ge 0 Furthermore for a isin (amid a]

983133 a

0

[GY (t)minusGX(t)] dt =

983133 a

a

[F (amid)minus F (t)] dt ge983133 a

a

[F (amid)minus F (t)] dt ge 0

where the last inequality follows from Jensenrsquos inequality because F is concave on [a a] We conclude that GX

second-order stochastically dominates GY If the type density is not decreasing on [a a] then second-order stochastic dominance does not hold but

Proposer is hurt by pruning (a a) if the expression in condition (1) is increasing on [a a]

11

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

by Melumad and Shibano (1991) and Alonso and Matouschek (2008) we also cover stochasticmechanisms3 That is we allow for mechanisms in which Vetoer may choose among lotteriesover actions We view such mechanisms as not only theoretically important but also relevantin applications For instance in the hiring application mentioned earlier the search committee(Proposer) can offer the organization (Vetoer) an option of hiring ldquothe best available candidatewith at least five yearsrsquo work experience in that countryrdquo Both parties would view this optionas a lottery over some subset of candidates

Stochastic mechanisms can sometimes be optimal in our framework Nevertheless weestablish that our sufficient conditions for full delegation no compromise and interval dele-gation (Propositions 1 2 and 3) ensure optimality of these (deterministic) mechanisms evenamong stochastic mechanisms Furthermore by permitting stochastic mechanisms our suffi-cient conditions are shown to also be necessary for a class of Proposerrsquos utility functions thatinclude linear and quadratic loss Our approach to handling stochastic mechanisms shouldbe useful in other delegation problems

Recently Kolotilin and Zapechelnyuk (2019) have introduced balanced delegation problemswhich are delegation problems in which certain extreme actions or outside options must beincluded Our setting fits into their general framework as one can assume the status quomust be part of the delegation set Kolotilin and Zapechelnyuk derive a general equivalencebetween such problems and monotone Bayesian persuasion problems More concretely theyshow how some results from the latter literature (eg Kolotilin 2018 Dworczak and Martini2019) can be brought to bear on ldquolinearrdquo balanced delegation problems4 Our approach ofdirectly studying the delegation problem is complementary and has some advantages Firstit permits insights absent said linearity this is most evident in our full delegation resultSecond we believe it provides some more transparent economic intuitions Third unlikeKolotilin and Zapechelnyuk (2019) we can address stochastic mechanisms and necessity ofour sufficient conditions At a broader level note that by contrast to us Kolotilin and Za-pechelnyuk (2019) highlight applications concerning expertise-based delegation (ie withstate-dependent delegator preferences) Zapechelnyuk (2019) applies their methodology toa quality certification problem that he shows maps into a delegation problem in which the

3 A qualification is appropriate both Amador et al (2006) and Amador and Bagwell (2013) allow for moneyburning identifying conditions under which optimal mechanisms do not employ that instrument see AmadorBagwell and Frankel (2018) as well Stochastic mechanisms are equivalent to money burning for certain prefer-ence specifications but in general they are not equivalent Ambrus and Egorov (2017) discuss settings in whichmoney burning can be optimal

4 This linearity requires that the utilities of Proposer and (all types of) Vetoer viewed as a function of theaction have the same curvature

4

delegatorrsquos preferences are state independent

Like us Amador and Bagwell (2019) directly analyze a delegation problem with an out-side option They focus on monopoly regulation (Baron and Myerson 1982) without trans-fers Unlike in our paper their delegatorrsquos preferences are state dependent and they do notconsider stochastic mechanisms or necessity

Outline The rest of the paper proceeds as follows Section 2 presents our model Section 3contains our main results on the conditions for optimality of full delegation no compromiseand more broadly interval delegation Section 4 develops comparative statics and makescomparisons with other mechanisms Section 5 discusses some applications Section 6 con-cludes All proofs are in the appendices

2 Model

We consider a classic bargaining problem between two players a proposer (he) and a vetoplayer (she) who jointly determine a policy outcome or action a isin R In a manner elaboratedbelow Proposer makes a proposal that Vetoer can either accept or reject If Vetoer rejects astatus-quo action is preserved we normalize the status quo to 0

We assume both players have single-peaked utilities Proposerrsquos utility is u(a) that is con-cave maximized uniquely at a = 1 (essentially a normalization) and twice differentiable atall a ∕= 15 Unless indicated explicitly we use lsquoconcaversquo lsquoincreasingrsquo etc to mean lsquoweaklyconcaversquo lsquoweakly increasingrsquo etc We will sometimes invoke a restriction to the followingsubclass of Proposer preferences which stipulates a convex combination of the widely-usedlinear and quadratic loss functions

Condition LQ For some γ isin [0 1]

u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Vetoerrsquos utility is represented by minusl(|v minus a|) where l(middot) is strictly increasing So her utilityis symmetric around the unique ideal point v For tractability we assume l(|vminus a|) = (vminus a)2A subset of our results will rely only on Vetoerrsquos ordinal preferences for which the choiceof quadratic loss entails no loss of generality given that Vetoerrsquos utility is symmetric around

5 Permitting a point of nondifferentiability allows the linear loss function u(a) = minus|1 minus a| When we writeuprime(1) subsequently it refers to the left-derivative when u is not differentiable at 1

5

her ideal point Specifically Vetoerrsquos ordinal preferences are sufficient when we consider onlydeterministic mechanisms

A key ingredient of our model is that v is Vetoerrsquos private information We accordinglyrefer to v as Vetoerrsquos type It is drawn from a cumulative distribution F whose support is aninterval [v v] where we permit v = minusinfin andor v = infin We assume F admits a continuouslydifferentiable density f and that f(middot) gt 0 on [0 1] All aspects of the environment except thetype v are common knowledge If v were common knowledge this model would reduce tothat of Romer and Rosenthal (1978)

Naturally it is in Proposerrsquos interests to elicit information from Vetoer about v For exam-ple they might engage in cheap talk communication (Matthews 1989) possibly over multiplerounds or Proposer might make sequential proposals and so on To circumvent issues aboutexactly how the bargaining ensues we take a mechanism design approach Following the rev-elation principle we consider direct revelation mechanisms hereafter simply mechanisms

A deterministic mechanism is described by a real-valued function α(v) which specifies theaction when Vetoerrsquos type is v and must satisfy the usual incentive compatibility (IC) andindividual rationality (IR) conditions IC requires that each type v prefers α(v) to α(vprime) for anyvprime ∕= v IR requires that each type v prefers α(v) to the status quo 0 Notice that any determin-istic mechanism is equivalent to the Proposer offering a (closed) menu or delegation set A sube Rand Vetoer choosing an action from A cup 0 We will also consider the more general classof stochastic mechanisms which specify probability distributions over actions for each Vetoertype with analogous IC and IR constraints to those aforementioned Stochastic mechanismsare theoretically important because the revelation principle does not justify focussing only ondeterministic mechanisms As noted in the introduction they may also be relevant for appli-cations A notable contribution of this paper is to establish conditions under which despitethe absence of transfers stochastic mechanisms cannot improve upon deterministic ones6

We highlight that our model is one of private values Vetoerrsquos type does not directly affectProposerrsquos preferences This is by way of contrast with the delegation literature initiated byHolmstrom (1984) in which a principal gives discretion to an agent because of the agentrsquosexpertise ie because they have interdependent preferences We could extend our modeland analysis to incorporate this expertise-based delegation or discretion aspect but one ofour main themes is that discretion will emerge even when that is absent

6 Remark 2 below explains why stochastic mechanisms can be optimal Example E1 in Appendix E elaboratesAlonso and Matouschek (2008 p 281) provide a related example in their framework without a veto option seealso Kovac and Mylovanov (2009 Section 4)

6

As mentioned in the introduction the mechanism design approach we take can be viewedas identifying an upper bound on Proposerrsquos welfare That said as also mentioned there wefind the implementation via delegation sets quite realistic in various contexts

21 Proposerrsquos Problem

We now formally define Proposerrsquos problem Let M(R) denote the set of Borel probabil-ity distributions on R7 and M0(R) be the subset of distributions with finite expectation andfinite variance Denote by δa the degenerate distribution that puts probability 1 on action a Astochastic mechanismmdashor simply a mechanism without qualificationmdashis a measurable func-tion m [v v] rarr M0(R) with m(v) being the probability distribution over actions for typev8 To reduce notation for any deterministic mechanism α [v v] rarr R we also denote themechanism v 983041rarr δα(v) by α For any integrable function g A rarr R let Em(v)[g(a)] denote theexpectation of g(a) when a has distribution m(v) We only consider mechanisms m for whichv 983041rarr Em(v)[a] is integrable Define the subset of mechanisms

S =983051m [v v] rarr M0(R) | m(0) = δ0 and forallv lt vprime Em(v)[a] le Em(vprime)[a]

983052

That is S consists of mechanisms in which type 0 gets the status quo and a higher type re-ceives a higher expected action The first requirement is implied by IR since Vetoer can al-ways choose the status quo The second is implied by IC since Vetoerrsquos utility minus(v minus a)2 isequivalently represented by av minus a22 singlecrossing difference in (a v) yields monotonicityof Em(v)[a] in v from standard arguments (elaborated in fn 9 below)

Proposerrsquos problem is

maxmisinS

983133Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046=

983133 v

0

Em(s)[a]ds forallv isin [v v] (IC-env)

As noted above it is without loss to restrict attention to mechanisms in S The constraint(IC-env) captures the additional content of IC beyond monotonicity via an analog of thestandard envelope formula9 Note that since IC requires that no type prefer type 0rsquos lottery

7 We endow M(R) with the topology of weak convergence and the corresponding Borel σ-algebra8 There is no loss in restricting attention to M0(R) instead of M(R) because no type would choose a lottery

with infinite mean or variance given that the status quo is available9 Formally using quadratic utility IC requires forallv vprime Em(v)[avminusa22] ge Em(vprime)[avminusa22] and IR requires forallv

Em(v)[avminus a22] ge 0 An IC mechanism m thus satisfies IR if and only if m(0) = δ0 It follows that m satisfies IC

7

over its own and type 0rsquos IR constraint requires that it receive action 0 (captured in S) everytypersquos IR constraint is implied by type 0rsquos An optimal mechanism is a solution to problem (P)

If we restricted attention to deterministic mechanisms the analogous problem for Pro-poser would be

maxαisinA

983133u(α(v)) dF (v) (D)

st vα(v)minus α(v)22 =

983133 v

0

α(s)ds

whereA = α [v v] rarr R | α(0) = 0 and α is increasing

Any deterministic mechanism α that is IC has a corresponding (closed) delegation setAα =

983126v α(v) Conversely any delegation set A has a corresponding deterministic mecha-

nism αA where αA(v) is the action in A cup 0 that type v prefers the most (with ties brokenin favor of Proposer) Note that αA satisfies IC and IR While our formal analysis works withmechanisms it is easier and more economically intuitive to describe our main results whichconcern certain deterministic mechanisms using delegation sets

We emphasize some terminology an optimal deterministic mechanism (or an optimal dele-gation set) is a solution to problem (D) But when we say that a deterministic mechanism (ordelegation set) is optimal we mean that it solves problem (P) ie no stochastic mechanism canstrictly improve on it

Remark 1 We capture veto power via a standard interim IR constraint An alternative as inCompte and Jehiel (2009) would be to allow Vetoer to exercise her veto even after the mech-anism determines an action This is stronger than just ex-post IR because it also strengthensthe IC constraint when type v mimics type vprime v may veto a different set of allocations than vprime

would and so the action distribution that v evaluates the deviation with is not m(vprime) Whichform of veto power is conceptually appropriate depends on the application But any IC andIR deterministic mechanism also satisfies the ex-post veto constraint Hence the sufficientconditions we provide below for optimality of delegation sets would remain sufficient ⋄

and IR if and only if m isin S and the envelope formula (IC-env) holds To confirm this let V(v) = Em(v)[avminusa22]Mechanism m is IC if and only if V(v) = maxvprime Em(vprime)[av minus a22] which holds if and only if V is convex andV(v) = V(0) +

983125 v

0Em(s)[a]ds (Milgrom and Segal 2002) Consequently m is IC and IR if and only if Em(v)[a] is

increasing in v (IC-env) holds and m(0) = δ0

8

22 Preliminary Observations

Consider delegation sets Notice first that there is no loss for Proposer in including hisideal action 1 in the delegation set for any Vetoer type v either it does not affect the chosenaction or it results in a preferable action Next there is no loss for Proposer in excludingactions outside [0 1] shrinking a delegation set A that contains 1 to A cap [0 1] only results ineach type choosing an action closer to 1 We have

Lemma 1 There is an optimal delegation set A satisfying 1 isin A sube [0 1]

It would also be without loss to assume that a delegation set contains the status quo 0 Wedonrsquot do so however because it is convenient to sometimes describe optimal delegation setswithout including 0

For any a isin (0 1) the delegation set A = [a 1] strictly dominates the singleton a becauseA results in preferable actions for Proposer when v gt a This simple observation highlightsthe significance of giving Vetoer discretion despite our model shutting down the expertise-based rationale that the literature initiated by Holmstrom (1984) has focused on

While Proposer always wants to include action 1 in the delegation set he faces a tradeoffwhen including any action a isin (0 1) Allowing Vetoer to choose such an action a reduces theprobability of a veto (or any action less than a) but also reduces the probability that Vetoerchooses an action even higher than a which Proposer would prefer to a

3 Delegation and Optimal Mechanisms

31 Full Delegation

In light of Lemma 1 we refer to the delegation set [0 1] as full delegation Note that full del-egation does impose some constraints on Vetoer But the constraints are minimal only actionsoutside the convex hull of the status quo and Proposerrsquos ideal point are excluded Given theveto-bargaining institution an outcome of full delegation starkly captures how Vetoerrsquos pri-vate information can corrode Proposerrsquos bargaining or agenda-setting power Indeed if thetype distributionrsquos support is [0 1] then Vetoer gets her first best Regardless of the supporthowever full delegation results in ex-post Pareto efficiency unlike in most other settings thatconfer information rents Full delegation thus contrasts sharply with the outcome under com-plete information (Romer and Rosenthal 1978) in which case Proposer would make Vetoerwith ideal point v isin (0 12] indifferent with exercising the veto while getting his own ideal

9

action 1 from types v isin (12 1) It also contrasts with the outcome under incomplete informa-tion were Proposer restricted to making a singleton proposal In that case the proposal wouldlead to a veto by some subinterval of types v isin [0 1] hence to ex-post Pareto inefficiency andall Vetoer types would be weakly worse off many strictly10

It is thus of interest to know when full delegation is optimal The following quantityconcerning the concavity of Proposerrsquos utility will play a key role in our analysis

κ = infaisin[01)

minusuprimeprime(a)

Proposition 1 (Full delegation) Full delegation is optimal if

κF (v)minus uprime(v)f(v) is increasing on [0 1] (1)

Conversely under Condition LQ full delegation is optimal only if (1) holds

Consider sufficiency in Proposition 1 If we had restricted attention to deterministic mech-anisms and assumed that Proposerrsquos utility is a quadratic loss function then the result wouldfollow from a result in Alonso and Matouschek (2008) even though their model does nothave a veto constraint and as such highlights expertise-based delegation To make the con-nection we observe that when u(a) = minus(1 minus a)2 condition (1) is equivalent to Alonso andMatouschekrsquos ldquobackward biasrdquo being convex on [0 1] Their Proposition 2 then implies thatif 0 1 is contained in the optimal delegation set then the interval [0 1] is contained in theoptimal delegation set But recall from Lemma 1 that in choosing among delegation sets Pro-poser need not offer any action outside [0 1] and can offer his ideal point 1 moreover he mayas well offer the status quo 0 It follows that full delegation is an optimal delegation set

We therefore emphasize that Proposition 1 establishes optimality among more generalProposer preferences and stochastic mechanisms11 The proposition directly implies

Corollary 1 Full delegation is optimal if the type density is decreasing on [0 1]

In particular it is sufficient that the type density is unimodal with a negative mode Toobtain intuition for the corollary consider removing any interval (a a) from a delegation set

10 Note that vetoes can have positive probability even under full delegation this is the case if and only ifProb(v lt 0) gt 0

11 In a model without an outside option Kovac and Mylovanov (2009) provide sufficient conditions for cer-tain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic lossfunction

10

A that contains [a a] This change induces Vetoer with type v isin (a a) to choose between a anda Due to her symmetric utility function Vetoer will choose a when v isin

983043a a+a

2

983044 which harms

Proposer while Vetoer will choose a when v isin983043a+a2 a983044 which benefits Proposer When the

type density is decreasing on [a a] the former possibility is more likely in fact the pruneddelegation set induces an action distribution that is second-order stochastically dominated12

Since Proposerrsquos utility is concave he prefers the original delegation set A As any (closed)delegation set can be obtained by successively removing open intervals full delegation is anoptimal delegation set While this explanation applies among delegation sets Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms

Removing an interval increases the expected action when the type density is increasingbut it also increases the probability of a lower action Thus when Proposer is risk averseit can be optimal to not remove an interval even if the density is increasing on that intervalThis explains why condition (1) is weaker than f decreasing on [0 1] In general removing aninterval is optimal only if the density is increasing quickly relative to Proposerrsquos risk aversionThis suggests that full delegation is optimal whenever Proposer is sufficiently risk averseProposition 1 allows us to formalize the point using the Arrow-Prat (Arrow 1965 Pratt 1964)coefficient of absolute risk aversion

Corollary 2 Full delegation is optimal if Proposer is sufficiently risk averse ie if infaisin[01)

minusuprimeprime(a)uprime(a)

is sufficiently large

Consider now necessity in Proposition 1 If Proposer has a linear loss utility then our pre-ceding discussion explains why a delegation set A containing [a a] sube [0 1] should be prunedto A (a a) if the type density is increasing on this interval the expected action increasesHence the converse of Corollary 1 holds for linear loss utility For quadratic loss utility (andwith additional smoothness assumptions) Alonso and Matouschekrsquos (2008) Proposition 2 im-plies that that a weaker condition F (v)minus (1minus v)f(v) increasing on [0 1] is necessary for full

12 Let GX denote the cumulative distribution of the action induced by A GY denote that induced by A (a a)and let amid = (a+ a)2 Since F is the distribution of v it holds that GX(a) = GY (a) for a isin (a a] GX(a) =F (a) on [a a] and GY (a) = F (amid) for a isin [a a) Consequently for any a isin [a amid] GY (a) ge GX(a) and983125 a

0[GY (t)minusGX(t)] dt ge 0 Furthermore for a isin (amid a]

983133 a

0

[GY (t)minusGX(t)] dt =

983133 a

a

[F (amid)minus F (t)] dt ge983133 a

a

[F (amid)minus F (t)] dt ge 0

where the last inequality follows from Jensenrsquos inequality because F is concave on [a a] We conclude that GX

second-order stochastically dominates GY If the type density is not decreasing on [a a] then second-order stochastic dominance does not hold but

Proposer is hurt by pruning (a a) if the expression in condition (1) is increasing on [a a]

11

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

delegatorrsquos preferences are state independent

Like us Amador and Bagwell (2019) directly analyze a delegation problem with an out-side option They focus on monopoly regulation (Baron and Myerson 1982) without trans-fers Unlike in our paper their delegatorrsquos preferences are state dependent and they do notconsider stochastic mechanisms or necessity

Outline The rest of the paper proceeds as follows Section 2 presents our model Section 3contains our main results on the conditions for optimality of full delegation no compromiseand more broadly interval delegation Section 4 develops comparative statics and makescomparisons with other mechanisms Section 5 discusses some applications Section 6 con-cludes All proofs are in the appendices

2 Model

We consider a classic bargaining problem between two players a proposer (he) and a vetoplayer (she) who jointly determine a policy outcome or action a isin R In a manner elaboratedbelow Proposer makes a proposal that Vetoer can either accept or reject If Vetoer rejects astatus-quo action is preserved we normalize the status quo to 0

We assume both players have single-peaked utilities Proposerrsquos utility is u(a) that is con-cave maximized uniquely at a = 1 (essentially a normalization) and twice differentiable atall a ∕= 15 Unless indicated explicitly we use lsquoconcaversquo lsquoincreasingrsquo etc to mean lsquoweaklyconcaversquo lsquoweakly increasingrsquo etc We will sometimes invoke a restriction to the followingsubclass of Proposer preferences which stipulates a convex combination of the widely-usedlinear and quadratic loss functions

Condition LQ For some γ isin [0 1]

u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Vetoerrsquos utility is represented by minusl(|v minus a|) where l(middot) is strictly increasing So her utilityis symmetric around the unique ideal point v For tractability we assume l(|vminus a|) = (vminus a)2A subset of our results will rely only on Vetoerrsquos ordinal preferences for which the choiceof quadratic loss entails no loss of generality given that Vetoerrsquos utility is symmetric around

5 Permitting a point of nondifferentiability allows the linear loss function u(a) = minus|1 minus a| When we writeuprime(1) subsequently it refers to the left-derivative when u is not differentiable at 1

5

her ideal point Specifically Vetoerrsquos ordinal preferences are sufficient when we consider onlydeterministic mechanisms

A key ingredient of our model is that v is Vetoerrsquos private information We accordinglyrefer to v as Vetoerrsquos type It is drawn from a cumulative distribution F whose support is aninterval [v v] where we permit v = minusinfin andor v = infin We assume F admits a continuouslydifferentiable density f and that f(middot) gt 0 on [0 1] All aspects of the environment except thetype v are common knowledge If v were common knowledge this model would reduce tothat of Romer and Rosenthal (1978)

Naturally it is in Proposerrsquos interests to elicit information from Vetoer about v For exam-ple they might engage in cheap talk communication (Matthews 1989) possibly over multiplerounds or Proposer might make sequential proposals and so on To circumvent issues aboutexactly how the bargaining ensues we take a mechanism design approach Following the rev-elation principle we consider direct revelation mechanisms hereafter simply mechanisms

A deterministic mechanism is described by a real-valued function α(v) which specifies theaction when Vetoerrsquos type is v and must satisfy the usual incentive compatibility (IC) andindividual rationality (IR) conditions IC requires that each type v prefers α(v) to α(vprime) for anyvprime ∕= v IR requires that each type v prefers α(v) to the status quo 0 Notice that any determin-istic mechanism is equivalent to the Proposer offering a (closed) menu or delegation set A sube Rand Vetoer choosing an action from A cup 0 We will also consider the more general classof stochastic mechanisms which specify probability distributions over actions for each Vetoertype with analogous IC and IR constraints to those aforementioned Stochastic mechanismsare theoretically important because the revelation principle does not justify focussing only ondeterministic mechanisms As noted in the introduction they may also be relevant for appli-cations A notable contribution of this paper is to establish conditions under which despitethe absence of transfers stochastic mechanisms cannot improve upon deterministic ones6

We highlight that our model is one of private values Vetoerrsquos type does not directly affectProposerrsquos preferences This is by way of contrast with the delegation literature initiated byHolmstrom (1984) in which a principal gives discretion to an agent because of the agentrsquosexpertise ie because they have interdependent preferences We could extend our modeland analysis to incorporate this expertise-based delegation or discretion aspect but one ofour main themes is that discretion will emerge even when that is absent

6 Remark 2 below explains why stochastic mechanisms can be optimal Example E1 in Appendix E elaboratesAlonso and Matouschek (2008 p 281) provide a related example in their framework without a veto option seealso Kovac and Mylovanov (2009 Section 4)

6

As mentioned in the introduction the mechanism design approach we take can be viewedas identifying an upper bound on Proposerrsquos welfare That said as also mentioned there wefind the implementation via delegation sets quite realistic in various contexts

21 Proposerrsquos Problem

We now formally define Proposerrsquos problem Let M(R) denote the set of Borel probabil-ity distributions on R7 and M0(R) be the subset of distributions with finite expectation andfinite variance Denote by δa the degenerate distribution that puts probability 1 on action a Astochastic mechanismmdashor simply a mechanism without qualificationmdashis a measurable func-tion m [v v] rarr M0(R) with m(v) being the probability distribution over actions for typev8 To reduce notation for any deterministic mechanism α [v v] rarr R we also denote themechanism v 983041rarr δα(v) by α For any integrable function g A rarr R let Em(v)[g(a)] denote theexpectation of g(a) when a has distribution m(v) We only consider mechanisms m for whichv 983041rarr Em(v)[a] is integrable Define the subset of mechanisms

S =983051m [v v] rarr M0(R) | m(0) = δ0 and forallv lt vprime Em(v)[a] le Em(vprime)[a]

983052

That is S consists of mechanisms in which type 0 gets the status quo and a higher type re-ceives a higher expected action The first requirement is implied by IR since Vetoer can al-ways choose the status quo The second is implied by IC since Vetoerrsquos utility minus(v minus a)2 isequivalently represented by av minus a22 singlecrossing difference in (a v) yields monotonicityof Em(v)[a] in v from standard arguments (elaborated in fn 9 below)

Proposerrsquos problem is

maxmisinS

983133Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046=

983133 v

0

Em(s)[a]ds forallv isin [v v] (IC-env)

As noted above it is without loss to restrict attention to mechanisms in S The constraint(IC-env) captures the additional content of IC beyond monotonicity via an analog of thestandard envelope formula9 Note that since IC requires that no type prefer type 0rsquos lottery

7 We endow M(R) with the topology of weak convergence and the corresponding Borel σ-algebra8 There is no loss in restricting attention to M0(R) instead of M(R) because no type would choose a lottery

with infinite mean or variance given that the status quo is available9 Formally using quadratic utility IC requires forallv vprime Em(v)[avminusa22] ge Em(vprime)[avminusa22] and IR requires forallv

Em(v)[avminus a22] ge 0 An IC mechanism m thus satisfies IR if and only if m(0) = δ0 It follows that m satisfies IC

7

over its own and type 0rsquos IR constraint requires that it receive action 0 (captured in S) everytypersquos IR constraint is implied by type 0rsquos An optimal mechanism is a solution to problem (P)

If we restricted attention to deterministic mechanisms the analogous problem for Pro-poser would be

maxαisinA

983133u(α(v)) dF (v) (D)

st vα(v)minus α(v)22 =

983133 v

0

α(s)ds

whereA = α [v v] rarr R | α(0) = 0 and α is increasing

Any deterministic mechanism α that is IC has a corresponding (closed) delegation setAα =

983126v α(v) Conversely any delegation set A has a corresponding deterministic mecha-

nism αA where αA(v) is the action in A cup 0 that type v prefers the most (with ties brokenin favor of Proposer) Note that αA satisfies IC and IR While our formal analysis works withmechanisms it is easier and more economically intuitive to describe our main results whichconcern certain deterministic mechanisms using delegation sets

We emphasize some terminology an optimal deterministic mechanism (or an optimal dele-gation set) is a solution to problem (D) But when we say that a deterministic mechanism (ordelegation set) is optimal we mean that it solves problem (P) ie no stochastic mechanism canstrictly improve on it

Remark 1 We capture veto power via a standard interim IR constraint An alternative as inCompte and Jehiel (2009) would be to allow Vetoer to exercise her veto even after the mech-anism determines an action This is stronger than just ex-post IR because it also strengthensthe IC constraint when type v mimics type vprime v may veto a different set of allocations than vprime

would and so the action distribution that v evaluates the deviation with is not m(vprime) Whichform of veto power is conceptually appropriate depends on the application But any IC andIR deterministic mechanism also satisfies the ex-post veto constraint Hence the sufficientconditions we provide below for optimality of delegation sets would remain sufficient ⋄

and IR if and only if m isin S and the envelope formula (IC-env) holds To confirm this let V(v) = Em(v)[avminusa22]Mechanism m is IC if and only if V(v) = maxvprime Em(vprime)[av minus a22] which holds if and only if V is convex andV(v) = V(0) +

983125 v

0Em(s)[a]ds (Milgrom and Segal 2002) Consequently m is IC and IR if and only if Em(v)[a] is

increasing in v (IC-env) holds and m(0) = δ0

8

22 Preliminary Observations

Consider delegation sets Notice first that there is no loss for Proposer in including hisideal action 1 in the delegation set for any Vetoer type v either it does not affect the chosenaction or it results in a preferable action Next there is no loss for Proposer in excludingactions outside [0 1] shrinking a delegation set A that contains 1 to A cap [0 1] only results ineach type choosing an action closer to 1 We have

Lemma 1 There is an optimal delegation set A satisfying 1 isin A sube [0 1]

It would also be without loss to assume that a delegation set contains the status quo 0 Wedonrsquot do so however because it is convenient to sometimes describe optimal delegation setswithout including 0

For any a isin (0 1) the delegation set A = [a 1] strictly dominates the singleton a becauseA results in preferable actions for Proposer when v gt a This simple observation highlightsthe significance of giving Vetoer discretion despite our model shutting down the expertise-based rationale that the literature initiated by Holmstrom (1984) has focused on

While Proposer always wants to include action 1 in the delegation set he faces a tradeoffwhen including any action a isin (0 1) Allowing Vetoer to choose such an action a reduces theprobability of a veto (or any action less than a) but also reduces the probability that Vetoerchooses an action even higher than a which Proposer would prefer to a

3 Delegation and Optimal Mechanisms

31 Full Delegation

In light of Lemma 1 we refer to the delegation set [0 1] as full delegation Note that full del-egation does impose some constraints on Vetoer But the constraints are minimal only actionsoutside the convex hull of the status quo and Proposerrsquos ideal point are excluded Given theveto-bargaining institution an outcome of full delegation starkly captures how Vetoerrsquos pri-vate information can corrode Proposerrsquos bargaining or agenda-setting power Indeed if thetype distributionrsquos support is [0 1] then Vetoer gets her first best Regardless of the supporthowever full delegation results in ex-post Pareto efficiency unlike in most other settings thatconfer information rents Full delegation thus contrasts sharply with the outcome under com-plete information (Romer and Rosenthal 1978) in which case Proposer would make Vetoerwith ideal point v isin (0 12] indifferent with exercising the veto while getting his own ideal

9

action 1 from types v isin (12 1) It also contrasts with the outcome under incomplete informa-tion were Proposer restricted to making a singleton proposal In that case the proposal wouldlead to a veto by some subinterval of types v isin [0 1] hence to ex-post Pareto inefficiency andall Vetoer types would be weakly worse off many strictly10

It is thus of interest to know when full delegation is optimal The following quantityconcerning the concavity of Proposerrsquos utility will play a key role in our analysis

κ = infaisin[01)

minusuprimeprime(a)

Proposition 1 (Full delegation) Full delegation is optimal if

κF (v)minus uprime(v)f(v) is increasing on [0 1] (1)

Conversely under Condition LQ full delegation is optimal only if (1) holds

Consider sufficiency in Proposition 1 If we had restricted attention to deterministic mech-anisms and assumed that Proposerrsquos utility is a quadratic loss function then the result wouldfollow from a result in Alonso and Matouschek (2008) even though their model does nothave a veto constraint and as such highlights expertise-based delegation To make the con-nection we observe that when u(a) = minus(1 minus a)2 condition (1) is equivalent to Alonso andMatouschekrsquos ldquobackward biasrdquo being convex on [0 1] Their Proposition 2 then implies thatif 0 1 is contained in the optimal delegation set then the interval [0 1] is contained in theoptimal delegation set But recall from Lemma 1 that in choosing among delegation sets Pro-poser need not offer any action outside [0 1] and can offer his ideal point 1 moreover he mayas well offer the status quo 0 It follows that full delegation is an optimal delegation set

We therefore emphasize that Proposition 1 establishes optimality among more generalProposer preferences and stochastic mechanisms11 The proposition directly implies

Corollary 1 Full delegation is optimal if the type density is decreasing on [0 1]

In particular it is sufficient that the type density is unimodal with a negative mode Toobtain intuition for the corollary consider removing any interval (a a) from a delegation set

10 Note that vetoes can have positive probability even under full delegation this is the case if and only ifProb(v lt 0) gt 0

11 In a model without an outside option Kovac and Mylovanov (2009) provide sufficient conditions for cer-tain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic lossfunction

10

A that contains [a a] This change induces Vetoer with type v isin (a a) to choose between a anda Due to her symmetric utility function Vetoer will choose a when v isin

983043a a+a

2

983044 which harms

Proposer while Vetoer will choose a when v isin983043a+a2 a983044 which benefits Proposer When the

type density is decreasing on [a a] the former possibility is more likely in fact the pruneddelegation set induces an action distribution that is second-order stochastically dominated12

Since Proposerrsquos utility is concave he prefers the original delegation set A As any (closed)delegation set can be obtained by successively removing open intervals full delegation is anoptimal delegation set While this explanation applies among delegation sets Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms

Removing an interval increases the expected action when the type density is increasingbut it also increases the probability of a lower action Thus when Proposer is risk averseit can be optimal to not remove an interval even if the density is increasing on that intervalThis explains why condition (1) is weaker than f decreasing on [0 1] In general removing aninterval is optimal only if the density is increasing quickly relative to Proposerrsquos risk aversionThis suggests that full delegation is optimal whenever Proposer is sufficiently risk averseProposition 1 allows us to formalize the point using the Arrow-Prat (Arrow 1965 Pratt 1964)coefficient of absolute risk aversion

Corollary 2 Full delegation is optimal if Proposer is sufficiently risk averse ie if infaisin[01)

minusuprimeprime(a)uprime(a)

is sufficiently large

Consider now necessity in Proposition 1 If Proposer has a linear loss utility then our pre-ceding discussion explains why a delegation set A containing [a a] sube [0 1] should be prunedto A (a a) if the type density is increasing on this interval the expected action increasesHence the converse of Corollary 1 holds for linear loss utility For quadratic loss utility (andwith additional smoothness assumptions) Alonso and Matouschekrsquos (2008) Proposition 2 im-plies that that a weaker condition F (v)minus (1minus v)f(v) increasing on [0 1] is necessary for full

12 Let GX denote the cumulative distribution of the action induced by A GY denote that induced by A (a a)and let amid = (a+ a)2 Since F is the distribution of v it holds that GX(a) = GY (a) for a isin (a a] GX(a) =F (a) on [a a] and GY (a) = F (amid) for a isin [a a) Consequently for any a isin [a amid] GY (a) ge GX(a) and983125 a

0[GY (t)minusGX(t)] dt ge 0 Furthermore for a isin (amid a]

983133 a

0

[GY (t)minusGX(t)] dt =

983133 a

a

[F (amid)minus F (t)] dt ge983133 a

a

[F (amid)minus F (t)] dt ge 0

where the last inequality follows from Jensenrsquos inequality because F is concave on [a a] We conclude that GX

second-order stochastically dominates GY If the type density is not decreasing on [a a] then second-order stochastic dominance does not hold but

Proposer is hurt by pruning (a a) if the expression in condition (1) is increasing on [a a]

11

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

her ideal point Specifically Vetoerrsquos ordinal preferences are sufficient when we consider onlydeterministic mechanisms

A key ingredient of our model is that v is Vetoerrsquos private information We accordinglyrefer to v as Vetoerrsquos type It is drawn from a cumulative distribution F whose support is aninterval [v v] where we permit v = minusinfin andor v = infin We assume F admits a continuouslydifferentiable density f and that f(middot) gt 0 on [0 1] All aspects of the environment except thetype v are common knowledge If v were common knowledge this model would reduce tothat of Romer and Rosenthal (1978)

Naturally it is in Proposerrsquos interests to elicit information from Vetoer about v For exam-ple they might engage in cheap talk communication (Matthews 1989) possibly over multiplerounds or Proposer might make sequential proposals and so on To circumvent issues aboutexactly how the bargaining ensues we take a mechanism design approach Following the rev-elation principle we consider direct revelation mechanisms hereafter simply mechanisms

A deterministic mechanism is described by a real-valued function α(v) which specifies theaction when Vetoerrsquos type is v and must satisfy the usual incentive compatibility (IC) andindividual rationality (IR) conditions IC requires that each type v prefers α(v) to α(vprime) for anyvprime ∕= v IR requires that each type v prefers α(v) to the status quo 0 Notice that any determin-istic mechanism is equivalent to the Proposer offering a (closed) menu or delegation set A sube Rand Vetoer choosing an action from A cup 0 We will also consider the more general classof stochastic mechanisms which specify probability distributions over actions for each Vetoertype with analogous IC and IR constraints to those aforementioned Stochastic mechanismsare theoretically important because the revelation principle does not justify focussing only ondeterministic mechanisms As noted in the introduction they may also be relevant for appli-cations A notable contribution of this paper is to establish conditions under which despitethe absence of transfers stochastic mechanisms cannot improve upon deterministic ones6

We highlight that our model is one of private values Vetoerrsquos type does not directly affectProposerrsquos preferences This is by way of contrast with the delegation literature initiated byHolmstrom (1984) in which a principal gives discretion to an agent because of the agentrsquosexpertise ie because they have interdependent preferences We could extend our modeland analysis to incorporate this expertise-based delegation or discretion aspect but one ofour main themes is that discretion will emerge even when that is absent

6 Remark 2 below explains why stochastic mechanisms can be optimal Example E1 in Appendix E elaboratesAlonso and Matouschek (2008 p 281) provide a related example in their framework without a veto option seealso Kovac and Mylovanov (2009 Section 4)

6

As mentioned in the introduction the mechanism design approach we take can be viewedas identifying an upper bound on Proposerrsquos welfare That said as also mentioned there wefind the implementation via delegation sets quite realistic in various contexts

21 Proposerrsquos Problem

We now formally define Proposerrsquos problem Let M(R) denote the set of Borel probabil-ity distributions on R7 and M0(R) be the subset of distributions with finite expectation andfinite variance Denote by δa the degenerate distribution that puts probability 1 on action a Astochastic mechanismmdashor simply a mechanism without qualificationmdashis a measurable func-tion m [v v] rarr M0(R) with m(v) being the probability distribution over actions for typev8 To reduce notation for any deterministic mechanism α [v v] rarr R we also denote themechanism v 983041rarr δα(v) by α For any integrable function g A rarr R let Em(v)[g(a)] denote theexpectation of g(a) when a has distribution m(v) We only consider mechanisms m for whichv 983041rarr Em(v)[a] is integrable Define the subset of mechanisms

S =983051m [v v] rarr M0(R) | m(0) = δ0 and forallv lt vprime Em(v)[a] le Em(vprime)[a]

983052

That is S consists of mechanisms in which type 0 gets the status quo and a higher type re-ceives a higher expected action The first requirement is implied by IR since Vetoer can al-ways choose the status quo The second is implied by IC since Vetoerrsquos utility minus(v minus a)2 isequivalently represented by av minus a22 singlecrossing difference in (a v) yields monotonicityof Em(v)[a] in v from standard arguments (elaborated in fn 9 below)

Proposerrsquos problem is

maxmisinS

983133Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046=

983133 v

0

Em(s)[a]ds forallv isin [v v] (IC-env)

As noted above it is without loss to restrict attention to mechanisms in S The constraint(IC-env) captures the additional content of IC beyond monotonicity via an analog of thestandard envelope formula9 Note that since IC requires that no type prefer type 0rsquos lottery

7 We endow M(R) with the topology of weak convergence and the corresponding Borel σ-algebra8 There is no loss in restricting attention to M0(R) instead of M(R) because no type would choose a lottery

with infinite mean or variance given that the status quo is available9 Formally using quadratic utility IC requires forallv vprime Em(v)[avminusa22] ge Em(vprime)[avminusa22] and IR requires forallv

Em(v)[avminus a22] ge 0 An IC mechanism m thus satisfies IR if and only if m(0) = δ0 It follows that m satisfies IC

7

over its own and type 0rsquos IR constraint requires that it receive action 0 (captured in S) everytypersquos IR constraint is implied by type 0rsquos An optimal mechanism is a solution to problem (P)

If we restricted attention to deterministic mechanisms the analogous problem for Pro-poser would be

maxαisinA

983133u(α(v)) dF (v) (D)

st vα(v)minus α(v)22 =

983133 v

0

α(s)ds

whereA = α [v v] rarr R | α(0) = 0 and α is increasing

Any deterministic mechanism α that is IC has a corresponding (closed) delegation setAα =

983126v α(v) Conversely any delegation set A has a corresponding deterministic mecha-

nism αA where αA(v) is the action in A cup 0 that type v prefers the most (with ties brokenin favor of Proposer) Note that αA satisfies IC and IR While our formal analysis works withmechanisms it is easier and more economically intuitive to describe our main results whichconcern certain deterministic mechanisms using delegation sets

We emphasize some terminology an optimal deterministic mechanism (or an optimal dele-gation set) is a solution to problem (D) But when we say that a deterministic mechanism (ordelegation set) is optimal we mean that it solves problem (P) ie no stochastic mechanism canstrictly improve on it

Remark 1 We capture veto power via a standard interim IR constraint An alternative as inCompte and Jehiel (2009) would be to allow Vetoer to exercise her veto even after the mech-anism determines an action This is stronger than just ex-post IR because it also strengthensthe IC constraint when type v mimics type vprime v may veto a different set of allocations than vprime

would and so the action distribution that v evaluates the deviation with is not m(vprime) Whichform of veto power is conceptually appropriate depends on the application But any IC andIR deterministic mechanism also satisfies the ex-post veto constraint Hence the sufficientconditions we provide below for optimality of delegation sets would remain sufficient ⋄

and IR if and only if m isin S and the envelope formula (IC-env) holds To confirm this let V(v) = Em(v)[avminusa22]Mechanism m is IC if and only if V(v) = maxvprime Em(vprime)[av minus a22] which holds if and only if V is convex andV(v) = V(0) +

983125 v

0Em(s)[a]ds (Milgrom and Segal 2002) Consequently m is IC and IR if and only if Em(v)[a] is

increasing in v (IC-env) holds and m(0) = δ0

8

22 Preliminary Observations

Consider delegation sets Notice first that there is no loss for Proposer in including hisideal action 1 in the delegation set for any Vetoer type v either it does not affect the chosenaction or it results in a preferable action Next there is no loss for Proposer in excludingactions outside [0 1] shrinking a delegation set A that contains 1 to A cap [0 1] only results ineach type choosing an action closer to 1 We have

Lemma 1 There is an optimal delegation set A satisfying 1 isin A sube [0 1]

It would also be without loss to assume that a delegation set contains the status quo 0 Wedonrsquot do so however because it is convenient to sometimes describe optimal delegation setswithout including 0

For any a isin (0 1) the delegation set A = [a 1] strictly dominates the singleton a becauseA results in preferable actions for Proposer when v gt a This simple observation highlightsthe significance of giving Vetoer discretion despite our model shutting down the expertise-based rationale that the literature initiated by Holmstrom (1984) has focused on

While Proposer always wants to include action 1 in the delegation set he faces a tradeoffwhen including any action a isin (0 1) Allowing Vetoer to choose such an action a reduces theprobability of a veto (or any action less than a) but also reduces the probability that Vetoerchooses an action even higher than a which Proposer would prefer to a

3 Delegation and Optimal Mechanisms

31 Full Delegation

In light of Lemma 1 we refer to the delegation set [0 1] as full delegation Note that full del-egation does impose some constraints on Vetoer But the constraints are minimal only actionsoutside the convex hull of the status quo and Proposerrsquos ideal point are excluded Given theveto-bargaining institution an outcome of full delegation starkly captures how Vetoerrsquos pri-vate information can corrode Proposerrsquos bargaining or agenda-setting power Indeed if thetype distributionrsquos support is [0 1] then Vetoer gets her first best Regardless of the supporthowever full delegation results in ex-post Pareto efficiency unlike in most other settings thatconfer information rents Full delegation thus contrasts sharply with the outcome under com-plete information (Romer and Rosenthal 1978) in which case Proposer would make Vetoerwith ideal point v isin (0 12] indifferent with exercising the veto while getting his own ideal

9

action 1 from types v isin (12 1) It also contrasts with the outcome under incomplete informa-tion were Proposer restricted to making a singleton proposal In that case the proposal wouldlead to a veto by some subinterval of types v isin [0 1] hence to ex-post Pareto inefficiency andall Vetoer types would be weakly worse off many strictly10

It is thus of interest to know when full delegation is optimal The following quantityconcerning the concavity of Proposerrsquos utility will play a key role in our analysis

κ = infaisin[01)

minusuprimeprime(a)

Proposition 1 (Full delegation) Full delegation is optimal if

κF (v)minus uprime(v)f(v) is increasing on [0 1] (1)

Conversely under Condition LQ full delegation is optimal only if (1) holds

Consider sufficiency in Proposition 1 If we had restricted attention to deterministic mech-anisms and assumed that Proposerrsquos utility is a quadratic loss function then the result wouldfollow from a result in Alonso and Matouschek (2008) even though their model does nothave a veto constraint and as such highlights expertise-based delegation To make the con-nection we observe that when u(a) = minus(1 minus a)2 condition (1) is equivalent to Alonso andMatouschekrsquos ldquobackward biasrdquo being convex on [0 1] Their Proposition 2 then implies thatif 0 1 is contained in the optimal delegation set then the interval [0 1] is contained in theoptimal delegation set But recall from Lemma 1 that in choosing among delegation sets Pro-poser need not offer any action outside [0 1] and can offer his ideal point 1 moreover he mayas well offer the status quo 0 It follows that full delegation is an optimal delegation set

We therefore emphasize that Proposition 1 establishes optimality among more generalProposer preferences and stochastic mechanisms11 The proposition directly implies

Corollary 1 Full delegation is optimal if the type density is decreasing on [0 1]

In particular it is sufficient that the type density is unimodal with a negative mode Toobtain intuition for the corollary consider removing any interval (a a) from a delegation set

10 Note that vetoes can have positive probability even under full delegation this is the case if and only ifProb(v lt 0) gt 0

11 In a model without an outside option Kovac and Mylovanov (2009) provide sufficient conditions for cer-tain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic lossfunction

10

A that contains [a a] This change induces Vetoer with type v isin (a a) to choose between a anda Due to her symmetric utility function Vetoer will choose a when v isin

983043a a+a

2

983044 which harms

Proposer while Vetoer will choose a when v isin983043a+a2 a983044 which benefits Proposer When the

type density is decreasing on [a a] the former possibility is more likely in fact the pruneddelegation set induces an action distribution that is second-order stochastically dominated12

Since Proposerrsquos utility is concave he prefers the original delegation set A As any (closed)delegation set can be obtained by successively removing open intervals full delegation is anoptimal delegation set While this explanation applies among delegation sets Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms

Removing an interval increases the expected action when the type density is increasingbut it also increases the probability of a lower action Thus when Proposer is risk averseit can be optimal to not remove an interval even if the density is increasing on that intervalThis explains why condition (1) is weaker than f decreasing on [0 1] In general removing aninterval is optimal only if the density is increasing quickly relative to Proposerrsquos risk aversionThis suggests that full delegation is optimal whenever Proposer is sufficiently risk averseProposition 1 allows us to formalize the point using the Arrow-Prat (Arrow 1965 Pratt 1964)coefficient of absolute risk aversion

Corollary 2 Full delegation is optimal if Proposer is sufficiently risk averse ie if infaisin[01)

minusuprimeprime(a)uprime(a)

is sufficiently large

Consider now necessity in Proposition 1 If Proposer has a linear loss utility then our pre-ceding discussion explains why a delegation set A containing [a a] sube [0 1] should be prunedto A (a a) if the type density is increasing on this interval the expected action increasesHence the converse of Corollary 1 holds for linear loss utility For quadratic loss utility (andwith additional smoothness assumptions) Alonso and Matouschekrsquos (2008) Proposition 2 im-plies that that a weaker condition F (v)minus (1minus v)f(v) increasing on [0 1] is necessary for full

12 Let GX denote the cumulative distribution of the action induced by A GY denote that induced by A (a a)and let amid = (a+ a)2 Since F is the distribution of v it holds that GX(a) = GY (a) for a isin (a a] GX(a) =F (a) on [a a] and GY (a) = F (amid) for a isin [a a) Consequently for any a isin [a amid] GY (a) ge GX(a) and983125 a

0[GY (t)minusGX(t)] dt ge 0 Furthermore for a isin (amid a]

983133 a

0

[GY (t)minusGX(t)] dt =

983133 a

a

[F (amid)minus F (t)] dt ge983133 a

a

[F (amid)minus F (t)] dt ge 0

where the last inequality follows from Jensenrsquos inequality because F is concave on [a a] We conclude that GX

second-order stochastically dominates GY If the type density is not decreasing on [a a] then second-order stochastic dominance does not hold but

Proposer is hurt by pruning (a a) if the expression in condition (1) is increasing on [a a]

11

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

As mentioned in the introduction the mechanism design approach we take can be viewedas identifying an upper bound on Proposerrsquos welfare That said as also mentioned there wefind the implementation via delegation sets quite realistic in various contexts

21 Proposerrsquos Problem

We now formally define Proposerrsquos problem Let M(R) denote the set of Borel probabil-ity distributions on R7 and M0(R) be the subset of distributions with finite expectation andfinite variance Denote by δa the degenerate distribution that puts probability 1 on action a Astochastic mechanismmdashor simply a mechanism without qualificationmdashis a measurable func-tion m [v v] rarr M0(R) with m(v) being the probability distribution over actions for typev8 To reduce notation for any deterministic mechanism α [v v] rarr R we also denote themechanism v 983041rarr δα(v) by α For any integrable function g A rarr R let Em(v)[g(a)] denote theexpectation of g(a) when a has distribution m(v) We only consider mechanisms m for whichv 983041rarr Em(v)[a] is integrable Define the subset of mechanisms

S =983051m [v v] rarr M0(R) | m(0) = δ0 and forallv lt vprime Em(v)[a] le Em(vprime)[a]

983052

That is S consists of mechanisms in which type 0 gets the status quo and a higher type re-ceives a higher expected action The first requirement is implied by IR since Vetoer can al-ways choose the status quo The second is implied by IC since Vetoerrsquos utility minus(v minus a)2 isequivalently represented by av minus a22 singlecrossing difference in (a v) yields monotonicityof Em(v)[a] in v from standard arguments (elaborated in fn 9 below)

Proposerrsquos problem is

maxmisinS

983133Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046=

983133 v

0

Em(s)[a]ds forallv isin [v v] (IC-env)

As noted above it is without loss to restrict attention to mechanisms in S The constraint(IC-env) captures the additional content of IC beyond monotonicity via an analog of thestandard envelope formula9 Note that since IC requires that no type prefer type 0rsquos lottery

7 We endow M(R) with the topology of weak convergence and the corresponding Borel σ-algebra8 There is no loss in restricting attention to M0(R) instead of M(R) because no type would choose a lottery

with infinite mean or variance given that the status quo is available9 Formally using quadratic utility IC requires forallv vprime Em(v)[avminusa22] ge Em(vprime)[avminusa22] and IR requires forallv

Em(v)[avminus a22] ge 0 An IC mechanism m thus satisfies IR if and only if m(0) = δ0 It follows that m satisfies IC

7

over its own and type 0rsquos IR constraint requires that it receive action 0 (captured in S) everytypersquos IR constraint is implied by type 0rsquos An optimal mechanism is a solution to problem (P)

If we restricted attention to deterministic mechanisms the analogous problem for Pro-poser would be

maxαisinA

983133u(α(v)) dF (v) (D)

st vα(v)minus α(v)22 =

983133 v

0

α(s)ds

whereA = α [v v] rarr R | α(0) = 0 and α is increasing

Any deterministic mechanism α that is IC has a corresponding (closed) delegation setAα =

983126v α(v) Conversely any delegation set A has a corresponding deterministic mecha-

nism αA where αA(v) is the action in A cup 0 that type v prefers the most (with ties brokenin favor of Proposer) Note that αA satisfies IC and IR While our formal analysis works withmechanisms it is easier and more economically intuitive to describe our main results whichconcern certain deterministic mechanisms using delegation sets

We emphasize some terminology an optimal deterministic mechanism (or an optimal dele-gation set) is a solution to problem (D) But when we say that a deterministic mechanism (ordelegation set) is optimal we mean that it solves problem (P) ie no stochastic mechanism canstrictly improve on it

Remark 1 We capture veto power via a standard interim IR constraint An alternative as inCompte and Jehiel (2009) would be to allow Vetoer to exercise her veto even after the mech-anism determines an action This is stronger than just ex-post IR because it also strengthensthe IC constraint when type v mimics type vprime v may veto a different set of allocations than vprime

would and so the action distribution that v evaluates the deviation with is not m(vprime) Whichform of veto power is conceptually appropriate depends on the application But any IC andIR deterministic mechanism also satisfies the ex-post veto constraint Hence the sufficientconditions we provide below for optimality of delegation sets would remain sufficient ⋄

and IR if and only if m isin S and the envelope formula (IC-env) holds To confirm this let V(v) = Em(v)[avminusa22]Mechanism m is IC if and only if V(v) = maxvprime Em(vprime)[av minus a22] which holds if and only if V is convex andV(v) = V(0) +

983125 v

0Em(s)[a]ds (Milgrom and Segal 2002) Consequently m is IC and IR if and only if Em(v)[a] is

increasing in v (IC-env) holds and m(0) = δ0

8

22 Preliminary Observations

Consider delegation sets Notice first that there is no loss for Proposer in including hisideal action 1 in the delegation set for any Vetoer type v either it does not affect the chosenaction or it results in a preferable action Next there is no loss for Proposer in excludingactions outside [0 1] shrinking a delegation set A that contains 1 to A cap [0 1] only results ineach type choosing an action closer to 1 We have

Lemma 1 There is an optimal delegation set A satisfying 1 isin A sube [0 1]

It would also be without loss to assume that a delegation set contains the status quo 0 Wedonrsquot do so however because it is convenient to sometimes describe optimal delegation setswithout including 0

For any a isin (0 1) the delegation set A = [a 1] strictly dominates the singleton a becauseA results in preferable actions for Proposer when v gt a This simple observation highlightsthe significance of giving Vetoer discretion despite our model shutting down the expertise-based rationale that the literature initiated by Holmstrom (1984) has focused on

While Proposer always wants to include action 1 in the delegation set he faces a tradeoffwhen including any action a isin (0 1) Allowing Vetoer to choose such an action a reduces theprobability of a veto (or any action less than a) but also reduces the probability that Vetoerchooses an action even higher than a which Proposer would prefer to a

3 Delegation and Optimal Mechanisms

31 Full Delegation

In light of Lemma 1 we refer to the delegation set [0 1] as full delegation Note that full del-egation does impose some constraints on Vetoer But the constraints are minimal only actionsoutside the convex hull of the status quo and Proposerrsquos ideal point are excluded Given theveto-bargaining institution an outcome of full delegation starkly captures how Vetoerrsquos pri-vate information can corrode Proposerrsquos bargaining or agenda-setting power Indeed if thetype distributionrsquos support is [0 1] then Vetoer gets her first best Regardless of the supporthowever full delegation results in ex-post Pareto efficiency unlike in most other settings thatconfer information rents Full delegation thus contrasts sharply with the outcome under com-plete information (Romer and Rosenthal 1978) in which case Proposer would make Vetoerwith ideal point v isin (0 12] indifferent with exercising the veto while getting his own ideal

9

action 1 from types v isin (12 1) It also contrasts with the outcome under incomplete informa-tion were Proposer restricted to making a singleton proposal In that case the proposal wouldlead to a veto by some subinterval of types v isin [0 1] hence to ex-post Pareto inefficiency andall Vetoer types would be weakly worse off many strictly10

It is thus of interest to know when full delegation is optimal The following quantityconcerning the concavity of Proposerrsquos utility will play a key role in our analysis

κ = infaisin[01)

minusuprimeprime(a)

Proposition 1 (Full delegation) Full delegation is optimal if

κF (v)minus uprime(v)f(v) is increasing on [0 1] (1)

Conversely under Condition LQ full delegation is optimal only if (1) holds

Consider sufficiency in Proposition 1 If we had restricted attention to deterministic mech-anisms and assumed that Proposerrsquos utility is a quadratic loss function then the result wouldfollow from a result in Alonso and Matouschek (2008) even though their model does nothave a veto constraint and as such highlights expertise-based delegation To make the con-nection we observe that when u(a) = minus(1 minus a)2 condition (1) is equivalent to Alonso andMatouschekrsquos ldquobackward biasrdquo being convex on [0 1] Their Proposition 2 then implies thatif 0 1 is contained in the optimal delegation set then the interval [0 1] is contained in theoptimal delegation set But recall from Lemma 1 that in choosing among delegation sets Pro-poser need not offer any action outside [0 1] and can offer his ideal point 1 moreover he mayas well offer the status quo 0 It follows that full delegation is an optimal delegation set

We therefore emphasize that Proposition 1 establishes optimality among more generalProposer preferences and stochastic mechanisms11 The proposition directly implies

Corollary 1 Full delegation is optimal if the type density is decreasing on [0 1]

In particular it is sufficient that the type density is unimodal with a negative mode Toobtain intuition for the corollary consider removing any interval (a a) from a delegation set

10 Note that vetoes can have positive probability even under full delegation this is the case if and only ifProb(v lt 0) gt 0

11 In a model without an outside option Kovac and Mylovanov (2009) provide sufficient conditions for cer-tain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic lossfunction

10

A that contains [a a] This change induces Vetoer with type v isin (a a) to choose between a anda Due to her symmetric utility function Vetoer will choose a when v isin

983043a a+a

2

983044 which harms

Proposer while Vetoer will choose a when v isin983043a+a2 a983044 which benefits Proposer When the

type density is decreasing on [a a] the former possibility is more likely in fact the pruneddelegation set induces an action distribution that is second-order stochastically dominated12

Since Proposerrsquos utility is concave he prefers the original delegation set A As any (closed)delegation set can be obtained by successively removing open intervals full delegation is anoptimal delegation set While this explanation applies among delegation sets Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms

Removing an interval increases the expected action when the type density is increasingbut it also increases the probability of a lower action Thus when Proposer is risk averseit can be optimal to not remove an interval even if the density is increasing on that intervalThis explains why condition (1) is weaker than f decreasing on [0 1] In general removing aninterval is optimal only if the density is increasing quickly relative to Proposerrsquos risk aversionThis suggests that full delegation is optimal whenever Proposer is sufficiently risk averseProposition 1 allows us to formalize the point using the Arrow-Prat (Arrow 1965 Pratt 1964)coefficient of absolute risk aversion

Corollary 2 Full delegation is optimal if Proposer is sufficiently risk averse ie if infaisin[01)

minusuprimeprime(a)uprime(a)

is sufficiently large

Consider now necessity in Proposition 1 If Proposer has a linear loss utility then our pre-ceding discussion explains why a delegation set A containing [a a] sube [0 1] should be prunedto A (a a) if the type density is increasing on this interval the expected action increasesHence the converse of Corollary 1 holds for linear loss utility For quadratic loss utility (andwith additional smoothness assumptions) Alonso and Matouschekrsquos (2008) Proposition 2 im-plies that that a weaker condition F (v)minus (1minus v)f(v) increasing on [0 1] is necessary for full

12 Let GX denote the cumulative distribution of the action induced by A GY denote that induced by A (a a)and let amid = (a+ a)2 Since F is the distribution of v it holds that GX(a) = GY (a) for a isin (a a] GX(a) =F (a) on [a a] and GY (a) = F (amid) for a isin [a a) Consequently for any a isin [a amid] GY (a) ge GX(a) and983125 a

0[GY (t)minusGX(t)] dt ge 0 Furthermore for a isin (amid a]

983133 a

0

[GY (t)minusGX(t)] dt =

983133 a

a

[F (amid)minus F (t)] dt ge983133 a

a

[F (amid)minus F (t)] dt ge 0

where the last inequality follows from Jensenrsquos inequality because F is concave on [a a] We conclude that GX

second-order stochastically dominates GY If the type density is not decreasing on [a a] then second-order stochastic dominance does not hold but

Proposer is hurt by pruning (a a) if the expression in condition (1) is increasing on [a a]

11

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

over its own and type 0rsquos IR constraint requires that it receive action 0 (captured in S) everytypersquos IR constraint is implied by type 0rsquos An optimal mechanism is a solution to problem (P)

If we restricted attention to deterministic mechanisms the analogous problem for Pro-poser would be

maxαisinA

983133u(α(v)) dF (v) (D)

st vα(v)minus α(v)22 =

983133 v

0

α(s)ds

whereA = α [v v] rarr R | α(0) = 0 and α is increasing

Any deterministic mechanism α that is IC has a corresponding (closed) delegation setAα =

983126v α(v) Conversely any delegation set A has a corresponding deterministic mecha-

nism αA where αA(v) is the action in A cup 0 that type v prefers the most (with ties brokenin favor of Proposer) Note that αA satisfies IC and IR While our formal analysis works withmechanisms it is easier and more economically intuitive to describe our main results whichconcern certain deterministic mechanisms using delegation sets

We emphasize some terminology an optimal deterministic mechanism (or an optimal dele-gation set) is a solution to problem (D) But when we say that a deterministic mechanism (ordelegation set) is optimal we mean that it solves problem (P) ie no stochastic mechanism canstrictly improve on it

Remark 1 We capture veto power via a standard interim IR constraint An alternative as inCompte and Jehiel (2009) would be to allow Vetoer to exercise her veto even after the mech-anism determines an action This is stronger than just ex-post IR because it also strengthensthe IC constraint when type v mimics type vprime v may veto a different set of allocations than vprime

would and so the action distribution that v evaluates the deviation with is not m(vprime) Whichform of veto power is conceptually appropriate depends on the application But any IC andIR deterministic mechanism also satisfies the ex-post veto constraint Hence the sufficientconditions we provide below for optimality of delegation sets would remain sufficient ⋄

and IR if and only if m isin S and the envelope formula (IC-env) holds To confirm this let V(v) = Em(v)[avminusa22]Mechanism m is IC if and only if V(v) = maxvprime Em(vprime)[av minus a22] which holds if and only if V is convex andV(v) = V(0) +

983125 v

0Em(s)[a]ds (Milgrom and Segal 2002) Consequently m is IC and IR if and only if Em(v)[a] is

increasing in v (IC-env) holds and m(0) = δ0

8

22 Preliminary Observations

Consider delegation sets Notice first that there is no loss for Proposer in including hisideal action 1 in the delegation set for any Vetoer type v either it does not affect the chosenaction or it results in a preferable action Next there is no loss for Proposer in excludingactions outside [0 1] shrinking a delegation set A that contains 1 to A cap [0 1] only results ineach type choosing an action closer to 1 We have

Lemma 1 There is an optimal delegation set A satisfying 1 isin A sube [0 1]

It would also be without loss to assume that a delegation set contains the status quo 0 Wedonrsquot do so however because it is convenient to sometimes describe optimal delegation setswithout including 0

For any a isin (0 1) the delegation set A = [a 1] strictly dominates the singleton a becauseA results in preferable actions for Proposer when v gt a This simple observation highlightsthe significance of giving Vetoer discretion despite our model shutting down the expertise-based rationale that the literature initiated by Holmstrom (1984) has focused on

While Proposer always wants to include action 1 in the delegation set he faces a tradeoffwhen including any action a isin (0 1) Allowing Vetoer to choose such an action a reduces theprobability of a veto (or any action less than a) but also reduces the probability that Vetoerchooses an action even higher than a which Proposer would prefer to a

3 Delegation and Optimal Mechanisms

31 Full Delegation

In light of Lemma 1 we refer to the delegation set [0 1] as full delegation Note that full del-egation does impose some constraints on Vetoer But the constraints are minimal only actionsoutside the convex hull of the status quo and Proposerrsquos ideal point are excluded Given theveto-bargaining institution an outcome of full delegation starkly captures how Vetoerrsquos pri-vate information can corrode Proposerrsquos bargaining or agenda-setting power Indeed if thetype distributionrsquos support is [0 1] then Vetoer gets her first best Regardless of the supporthowever full delegation results in ex-post Pareto efficiency unlike in most other settings thatconfer information rents Full delegation thus contrasts sharply with the outcome under com-plete information (Romer and Rosenthal 1978) in which case Proposer would make Vetoerwith ideal point v isin (0 12] indifferent with exercising the veto while getting his own ideal

9

action 1 from types v isin (12 1) It also contrasts with the outcome under incomplete informa-tion were Proposer restricted to making a singleton proposal In that case the proposal wouldlead to a veto by some subinterval of types v isin [0 1] hence to ex-post Pareto inefficiency andall Vetoer types would be weakly worse off many strictly10

It is thus of interest to know when full delegation is optimal The following quantityconcerning the concavity of Proposerrsquos utility will play a key role in our analysis

κ = infaisin[01)

minusuprimeprime(a)

Proposition 1 (Full delegation) Full delegation is optimal if

κF (v)minus uprime(v)f(v) is increasing on [0 1] (1)

Conversely under Condition LQ full delegation is optimal only if (1) holds

Consider sufficiency in Proposition 1 If we had restricted attention to deterministic mech-anisms and assumed that Proposerrsquos utility is a quadratic loss function then the result wouldfollow from a result in Alonso and Matouschek (2008) even though their model does nothave a veto constraint and as such highlights expertise-based delegation To make the con-nection we observe that when u(a) = minus(1 minus a)2 condition (1) is equivalent to Alonso andMatouschekrsquos ldquobackward biasrdquo being convex on [0 1] Their Proposition 2 then implies thatif 0 1 is contained in the optimal delegation set then the interval [0 1] is contained in theoptimal delegation set But recall from Lemma 1 that in choosing among delegation sets Pro-poser need not offer any action outside [0 1] and can offer his ideal point 1 moreover he mayas well offer the status quo 0 It follows that full delegation is an optimal delegation set

We therefore emphasize that Proposition 1 establishes optimality among more generalProposer preferences and stochastic mechanisms11 The proposition directly implies

Corollary 1 Full delegation is optimal if the type density is decreasing on [0 1]

In particular it is sufficient that the type density is unimodal with a negative mode Toobtain intuition for the corollary consider removing any interval (a a) from a delegation set

10 Note that vetoes can have positive probability even under full delegation this is the case if and only ifProb(v lt 0) gt 0

11 In a model without an outside option Kovac and Mylovanov (2009) provide sufficient conditions for cer-tain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic lossfunction

10

A that contains [a a] This change induces Vetoer with type v isin (a a) to choose between a anda Due to her symmetric utility function Vetoer will choose a when v isin

983043a a+a

2

983044 which harms

Proposer while Vetoer will choose a when v isin983043a+a2 a983044 which benefits Proposer When the

type density is decreasing on [a a] the former possibility is more likely in fact the pruneddelegation set induces an action distribution that is second-order stochastically dominated12

Since Proposerrsquos utility is concave he prefers the original delegation set A As any (closed)delegation set can be obtained by successively removing open intervals full delegation is anoptimal delegation set While this explanation applies among delegation sets Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms

Removing an interval increases the expected action when the type density is increasingbut it also increases the probability of a lower action Thus when Proposer is risk averseit can be optimal to not remove an interval even if the density is increasing on that intervalThis explains why condition (1) is weaker than f decreasing on [0 1] In general removing aninterval is optimal only if the density is increasing quickly relative to Proposerrsquos risk aversionThis suggests that full delegation is optimal whenever Proposer is sufficiently risk averseProposition 1 allows us to formalize the point using the Arrow-Prat (Arrow 1965 Pratt 1964)coefficient of absolute risk aversion

Corollary 2 Full delegation is optimal if Proposer is sufficiently risk averse ie if infaisin[01)

minusuprimeprime(a)uprime(a)

is sufficiently large

Consider now necessity in Proposition 1 If Proposer has a linear loss utility then our pre-ceding discussion explains why a delegation set A containing [a a] sube [0 1] should be prunedto A (a a) if the type density is increasing on this interval the expected action increasesHence the converse of Corollary 1 holds for linear loss utility For quadratic loss utility (andwith additional smoothness assumptions) Alonso and Matouschekrsquos (2008) Proposition 2 im-plies that that a weaker condition F (v)minus (1minus v)f(v) increasing on [0 1] is necessary for full

12 Let GX denote the cumulative distribution of the action induced by A GY denote that induced by A (a a)and let amid = (a+ a)2 Since F is the distribution of v it holds that GX(a) = GY (a) for a isin (a a] GX(a) =F (a) on [a a] and GY (a) = F (amid) for a isin [a a) Consequently for any a isin [a amid] GY (a) ge GX(a) and983125 a

0[GY (t)minusGX(t)] dt ge 0 Furthermore for a isin (amid a]

983133 a

0

[GY (t)minusGX(t)] dt =

983133 a

a

[F (amid)minus F (t)] dt ge983133 a

a

[F (amid)minus F (t)] dt ge 0

where the last inequality follows from Jensenrsquos inequality because F is concave on [a a] We conclude that GX

second-order stochastically dominates GY If the type density is not decreasing on [a a] then second-order stochastic dominance does not hold but

Proposer is hurt by pruning (a a) if the expression in condition (1) is increasing on [a a]

11

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

22 Preliminary Observations

Consider delegation sets Notice first that there is no loss for Proposer in including hisideal action 1 in the delegation set for any Vetoer type v either it does not affect the chosenaction or it results in a preferable action Next there is no loss for Proposer in excludingactions outside [0 1] shrinking a delegation set A that contains 1 to A cap [0 1] only results ineach type choosing an action closer to 1 We have

Lemma 1 There is an optimal delegation set A satisfying 1 isin A sube [0 1]

It would also be without loss to assume that a delegation set contains the status quo 0 Wedonrsquot do so however because it is convenient to sometimes describe optimal delegation setswithout including 0

For any a isin (0 1) the delegation set A = [a 1] strictly dominates the singleton a becauseA results in preferable actions for Proposer when v gt a This simple observation highlightsthe significance of giving Vetoer discretion despite our model shutting down the expertise-based rationale that the literature initiated by Holmstrom (1984) has focused on

While Proposer always wants to include action 1 in the delegation set he faces a tradeoffwhen including any action a isin (0 1) Allowing Vetoer to choose such an action a reduces theprobability of a veto (or any action less than a) but also reduces the probability that Vetoerchooses an action even higher than a which Proposer would prefer to a

3 Delegation and Optimal Mechanisms

31 Full Delegation

In light of Lemma 1 we refer to the delegation set [0 1] as full delegation Note that full del-egation does impose some constraints on Vetoer But the constraints are minimal only actionsoutside the convex hull of the status quo and Proposerrsquos ideal point are excluded Given theveto-bargaining institution an outcome of full delegation starkly captures how Vetoerrsquos pri-vate information can corrode Proposerrsquos bargaining or agenda-setting power Indeed if thetype distributionrsquos support is [0 1] then Vetoer gets her first best Regardless of the supporthowever full delegation results in ex-post Pareto efficiency unlike in most other settings thatconfer information rents Full delegation thus contrasts sharply with the outcome under com-plete information (Romer and Rosenthal 1978) in which case Proposer would make Vetoerwith ideal point v isin (0 12] indifferent with exercising the veto while getting his own ideal

9

action 1 from types v isin (12 1) It also contrasts with the outcome under incomplete informa-tion were Proposer restricted to making a singleton proposal In that case the proposal wouldlead to a veto by some subinterval of types v isin [0 1] hence to ex-post Pareto inefficiency andall Vetoer types would be weakly worse off many strictly10

It is thus of interest to know when full delegation is optimal The following quantityconcerning the concavity of Proposerrsquos utility will play a key role in our analysis

κ = infaisin[01)

minusuprimeprime(a)

Proposition 1 (Full delegation) Full delegation is optimal if

κF (v)minus uprime(v)f(v) is increasing on [0 1] (1)

Conversely under Condition LQ full delegation is optimal only if (1) holds

Consider sufficiency in Proposition 1 If we had restricted attention to deterministic mech-anisms and assumed that Proposerrsquos utility is a quadratic loss function then the result wouldfollow from a result in Alonso and Matouschek (2008) even though their model does nothave a veto constraint and as such highlights expertise-based delegation To make the con-nection we observe that when u(a) = minus(1 minus a)2 condition (1) is equivalent to Alonso andMatouschekrsquos ldquobackward biasrdquo being convex on [0 1] Their Proposition 2 then implies thatif 0 1 is contained in the optimal delegation set then the interval [0 1] is contained in theoptimal delegation set But recall from Lemma 1 that in choosing among delegation sets Pro-poser need not offer any action outside [0 1] and can offer his ideal point 1 moreover he mayas well offer the status quo 0 It follows that full delegation is an optimal delegation set

We therefore emphasize that Proposition 1 establishes optimality among more generalProposer preferences and stochastic mechanisms11 The proposition directly implies

Corollary 1 Full delegation is optimal if the type density is decreasing on [0 1]

In particular it is sufficient that the type density is unimodal with a negative mode Toobtain intuition for the corollary consider removing any interval (a a) from a delegation set

10 Note that vetoes can have positive probability even under full delegation this is the case if and only ifProb(v lt 0) gt 0

11 In a model without an outside option Kovac and Mylovanov (2009) provide sufficient conditions for cer-tain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic lossfunction

10

A that contains [a a] This change induces Vetoer with type v isin (a a) to choose between a anda Due to her symmetric utility function Vetoer will choose a when v isin

983043a a+a

2

983044 which harms

Proposer while Vetoer will choose a when v isin983043a+a2 a983044 which benefits Proposer When the

type density is decreasing on [a a] the former possibility is more likely in fact the pruneddelegation set induces an action distribution that is second-order stochastically dominated12

Since Proposerrsquos utility is concave he prefers the original delegation set A As any (closed)delegation set can be obtained by successively removing open intervals full delegation is anoptimal delegation set While this explanation applies among delegation sets Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms

Removing an interval increases the expected action when the type density is increasingbut it also increases the probability of a lower action Thus when Proposer is risk averseit can be optimal to not remove an interval even if the density is increasing on that intervalThis explains why condition (1) is weaker than f decreasing on [0 1] In general removing aninterval is optimal only if the density is increasing quickly relative to Proposerrsquos risk aversionThis suggests that full delegation is optimal whenever Proposer is sufficiently risk averseProposition 1 allows us to formalize the point using the Arrow-Prat (Arrow 1965 Pratt 1964)coefficient of absolute risk aversion

Corollary 2 Full delegation is optimal if Proposer is sufficiently risk averse ie if infaisin[01)

minusuprimeprime(a)uprime(a)

is sufficiently large

Consider now necessity in Proposition 1 If Proposer has a linear loss utility then our pre-ceding discussion explains why a delegation set A containing [a a] sube [0 1] should be prunedto A (a a) if the type density is increasing on this interval the expected action increasesHence the converse of Corollary 1 holds for linear loss utility For quadratic loss utility (andwith additional smoothness assumptions) Alonso and Matouschekrsquos (2008) Proposition 2 im-plies that that a weaker condition F (v)minus (1minus v)f(v) increasing on [0 1] is necessary for full

12 Let GX denote the cumulative distribution of the action induced by A GY denote that induced by A (a a)and let amid = (a+ a)2 Since F is the distribution of v it holds that GX(a) = GY (a) for a isin (a a] GX(a) =F (a) on [a a] and GY (a) = F (amid) for a isin [a a) Consequently for any a isin [a amid] GY (a) ge GX(a) and983125 a

0[GY (t)minusGX(t)] dt ge 0 Furthermore for a isin (amid a]

983133 a

0

[GY (t)minusGX(t)] dt =

983133 a

a

[F (amid)minus F (t)] dt ge983133 a

a

[F (amid)minus F (t)] dt ge 0

where the last inequality follows from Jensenrsquos inequality because F is concave on [a a] We conclude that GX

second-order stochastically dominates GY If the type density is not decreasing on [a a] then second-order stochastic dominance does not hold but

Proposer is hurt by pruning (a a) if the expression in condition (1) is increasing on [a a]

11

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

action 1 from types v isin (12 1) It also contrasts with the outcome under incomplete informa-tion were Proposer restricted to making a singleton proposal In that case the proposal wouldlead to a veto by some subinterval of types v isin [0 1] hence to ex-post Pareto inefficiency andall Vetoer types would be weakly worse off many strictly10

It is thus of interest to know when full delegation is optimal The following quantityconcerning the concavity of Proposerrsquos utility will play a key role in our analysis

κ = infaisin[01)

minusuprimeprime(a)

Proposition 1 (Full delegation) Full delegation is optimal if

κF (v)minus uprime(v)f(v) is increasing on [0 1] (1)

Conversely under Condition LQ full delegation is optimal only if (1) holds

Consider sufficiency in Proposition 1 If we had restricted attention to deterministic mech-anisms and assumed that Proposerrsquos utility is a quadratic loss function then the result wouldfollow from a result in Alonso and Matouschek (2008) even though their model does nothave a veto constraint and as such highlights expertise-based delegation To make the con-nection we observe that when u(a) = minus(1 minus a)2 condition (1) is equivalent to Alonso andMatouschekrsquos ldquobackward biasrdquo being convex on [0 1] Their Proposition 2 then implies thatif 0 1 is contained in the optimal delegation set then the interval [0 1] is contained in theoptimal delegation set But recall from Lemma 1 that in choosing among delegation sets Pro-poser need not offer any action outside [0 1] and can offer his ideal point 1 moreover he mayas well offer the status quo 0 It follows that full delegation is an optimal delegation set

We therefore emphasize that Proposition 1 establishes optimality among more generalProposer preferences and stochastic mechanisms11 The proposition directly implies

Corollary 1 Full delegation is optimal if the type density is decreasing on [0 1]

In particular it is sufficient that the type density is unimodal with a negative mode Toobtain intuition for the corollary consider removing any interval (a a) from a delegation set

10 Note that vetoes can have positive probability even under full delegation this is the case if and only ifProb(v lt 0) gt 0

11 In a model without an outside option Kovac and Mylovanov (2009) provide sufficient conditions for cer-tain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic lossfunction

10

A that contains [a a] This change induces Vetoer with type v isin (a a) to choose between a anda Due to her symmetric utility function Vetoer will choose a when v isin

983043a a+a

2

983044 which harms

Proposer while Vetoer will choose a when v isin983043a+a2 a983044 which benefits Proposer When the

type density is decreasing on [a a] the former possibility is more likely in fact the pruneddelegation set induces an action distribution that is second-order stochastically dominated12

Since Proposerrsquos utility is concave he prefers the original delegation set A As any (closed)delegation set can be obtained by successively removing open intervals full delegation is anoptimal delegation set While this explanation applies among delegation sets Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms

Removing an interval increases the expected action when the type density is increasingbut it also increases the probability of a lower action Thus when Proposer is risk averseit can be optimal to not remove an interval even if the density is increasing on that intervalThis explains why condition (1) is weaker than f decreasing on [0 1] In general removing aninterval is optimal only if the density is increasing quickly relative to Proposerrsquos risk aversionThis suggests that full delegation is optimal whenever Proposer is sufficiently risk averseProposition 1 allows us to formalize the point using the Arrow-Prat (Arrow 1965 Pratt 1964)coefficient of absolute risk aversion

Corollary 2 Full delegation is optimal if Proposer is sufficiently risk averse ie if infaisin[01)

minusuprimeprime(a)uprime(a)

is sufficiently large

Consider now necessity in Proposition 1 If Proposer has a linear loss utility then our pre-ceding discussion explains why a delegation set A containing [a a] sube [0 1] should be prunedto A (a a) if the type density is increasing on this interval the expected action increasesHence the converse of Corollary 1 holds for linear loss utility For quadratic loss utility (andwith additional smoothness assumptions) Alonso and Matouschekrsquos (2008) Proposition 2 im-plies that that a weaker condition F (v)minus (1minus v)f(v) increasing on [0 1] is necessary for full

12 Let GX denote the cumulative distribution of the action induced by A GY denote that induced by A (a a)and let amid = (a+ a)2 Since F is the distribution of v it holds that GX(a) = GY (a) for a isin (a a] GX(a) =F (a) on [a a] and GY (a) = F (amid) for a isin [a a) Consequently for any a isin [a amid] GY (a) ge GX(a) and983125 a

0[GY (t)minusGX(t)] dt ge 0 Furthermore for a isin (amid a]

983133 a

0

[GY (t)minusGX(t)] dt =

983133 a

a

[F (amid)minus F (t)] dt ge983133 a

a

[F (amid)minus F (t)] dt ge 0

where the last inequality follows from Jensenrsquos inequality because F is concave on [a a] We conclude that GX

second-order stochastically dominates GY If the type density is not decreasing on [a a] then second-order stochastic dominance does not hold but

Proposer is hurt by pruning (a a) if the expression in condition (1) is increasing on [a a]

11

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

A that contains [a a] This change induces Vetoer with type v isin (a a) to choose between a anda Due to her symmetric utility function Vetoer will choose a when v isin

983043a a+a

2

983044 which harms

Proposer while Vetoer will choose a when v isin983043a+a2 a983044 which benefits Proposer When the

type density is decreasing on [a a] the former possibility is more likely in fact the pruneddelegation set induces an action distribution that is second-order stochastically dominated12

Since Proposerrsquos utility is concave he prefers the original delegation set A As any (closed)delegation set can be obtained by successively removing open intervals full delegation is anoptimal delegation set While this explanation applies among delegation sets Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms

Removing an interval increases the expected action when the type density is increasingbut it also increases the probability of a lower action Thus when Proposer is risk averseit can be optimal to not remove an interval even if the density is increasing on that intervalThis explains why condition (1) is weaker than f decreasing on [0 1] In general removing aninterval is optimal only if the density is increasing quickly relative to Proposerrsquos risk aversionThis suggests that full delegation is optimal whenever Proposer is sufficiently risk averseProposition 1 allows us to formalize the point using the Arrow-Prat (Arrow 1965 Pratt 1964)coefficient of absolute risk aversion

Corollary 2 Full delegation is optimal if Proposer is sufficiently risk averse ie if infaisin[01)

minusuprimeprime(a)uprime(a)

is sufficiently large

Consider now necessity in Proposition 1 If Proposer has a linear loss utility then our pre-ceding discussion explains why a delegation set A containing [a a] sube [0 1] should be prunedto A (a a) if the type density is increasing on this interval the expected action increasesHence the converse of Corollary 1 holds for linear loss utility For quadratic loss utility (andwith additional smoothness assumptions) Alonso and Matouschekrsquos (2008) Proposition 2 im-plies that that a weaker condition F (v)minus (1minus v)f(v) increasing on [0 1] is necessary for full

12 Let GX denote the cumulative distribution of the action induced by A GY denote that induced by A (a a)and let amid = (a+ a)2 Since F is the distribution of v it holds that GX(a) = GY (a) for a isin (a a] GX(a) =F (a) on [a a] and GY (a) = F (amid) for a isin [a a) Consequently for any a isin [a amid] GY (a) ge GX(a) and983125 a

0[GY (t)minusGX(t)] dt ge 0 Furthermore for a isin (amid a]

983133 a

0

[GY (t)minusGX(t)] dt =

983133 a

a

[F (amid)minus F (t)] dt ge983133 a

a

[F (amid)minus F (t)] dt ge 0

where the last inequality follows from Jensenrsquos inequality because F is concave on [a a] We conclude that GX

second-order stochastically dominates GY If the type density is not decreasing on [a a] then second-order stochastic dominance does not hold but

Proposer is hurt by pruning (a a) if the expression in condition (1) is increasing on [a a]

11

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

delegation to be optimal Proposition 1 subsumes these two cases by deducing necessity ofcondition (1) for the linear-quadratic family of utilities (Condition LQ)13

We note that were Vetoerrsquos lowest type v isin (0 1) contrary to our assumption that v le 0then full delegationmdashor even the interval [v 1]mdashwould never be an optimal delegation setProposer would not offer any action below min2v 1

32 No Compromise

The other extreme from full delegation is no compromise Proposer makes a take-it-or-leave-it offer of his own ideal action not offering any other action Of course Vetoer can choose thestatus quo as well When Proposer has a linear loss utilitymdashor a fortiori if we had permittedu(a) to be convex on [0 1]mdashthen no compromise is an optimal delegation set whenever thetype density f is increasing on [0 1] This follows from reversing the previous subsectionrsquossecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing But neither is linear loss utility nor convexity of F on [0 1] required for optimalityof no compromise

Proposition 2 Assume Condition LQ No compromise is optimal if and only if

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus sfor all 1 ge t gt 12 gt s ge 0

Under linear loss utility (so uprime(1) = uprime(0) = 1 and κ = 0) the condition in Proposition 2simplifies to f(12) being a subgradient of F at 12 on the domain [0 1] This subgradientcondition is weaker than F being convex on [0 1]

Remark 2 With linear loss utility no compromise can be an optimal delegation set (ie de-terministic mechanism) even if the subgradient condition does not hold However there willthen be a stochastic mechanism that Proposer strictly prefers This situation can arise forexample when the type density is strictly increasing except on a small interval around 12where it is strictly decreasing Intuitively Proposer would like to delegate a small set ofactions around 12 to types close to 12 but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above 12 choosing an action closeto 12 rather than 1 By contrast lotteries with expected value 12 can be used to attract onlytypes close to 12 Example E1 in Appendix E elaborates ⋄

13 Following our general methodology discussed in Subsection 34 our proof of necessity uses the availabilityof stochastic mechanisms But we can establish that under Condition LQ (1) is necessary even for full delegationto be an optimal delegation set

12

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

We also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ In particular it can be shown that if no compromise is an optimaldelegation set for some u then it is also an optimal delegation set for any utility function thatis a convex transformation of u

On the other hand no compromise is not optimalmdashnot even an optimal delegation setmdashifProposerrsquos utility is differentiable at his ideal point a = 1 (which implies uprime(1) = 0)14 Thereason is that when uprime(1) = 0 Proposer would strictly benefit from offering a small interval[1minusε 1] or even just the action 1minusε instead of only offering action 1 For Proposerrsquos decreasein utility from getting an action slightly lower than 1 is second order but there is a first-orderincrease in the probability of avoiding a veto

33 Interval Delegation

Both full delegation and no compromise are special cases of interval delegation Proposeroffers an interval and Vetoer chooses an action from either that interval or the status quo Itfollows from Lemma 1 that when interval delegation is optimal there is always an optimalinterval of the form [c 1] for some c isin [0 1] One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option implicitly the maximal acceptable option isProposerrsquos ideal point Interval delegation without a status quo has been a central focus ofthe prior literature intervals are simple tractable and lend themselves to comparative staticsArguably intervals also map more naturally into proposals likely to emerge in applications

Proposition 3 The interval delegation set [clowast 1] with clowast isin [0 1] is optimal if

(i) κF (v)minus uprime(v)f(v) is increasing on [clowast 1]

(ii) (uprime(clowast) + κ(clowast minus t)) F (t)minusF (clowast2)tminusclowast2 ge uprime(clowast)F (clowast)minusF (clowast2)

clowast2 for all t isin (clowast2 clowast] and

(iii) uprime(clowast)F (clowast)minusF (clowast2)clowast2 ge (uprime(0)minus κs) F (clowast2)minusF (s)

clowast2minussfor all s isin [0 clowast2)

Conversely under Condition LQ the delegation set [clowast 1] with clowast isin (0 1) is optimal only if conditions(i) (ii) and (iii) above hold

We discuss sufficiency The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1 it ensures that there is no benefit to not fully delegating theinterval [clowast 1] taking as given that Vetoer can choose clowast For linear loss utility the condition

14 Note that when uprime(1) = 0 the condition in Proposition 2 fails its left-hand side is 0 when t = 1 while itsright-hand side is strictly positive when s = 0

13

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

v

F (v)

clowast2 clowast0 1

1

Figure 1 ndash Conditions (i)mdash(iii) of Proposition 3 for linear loss utility F is concave on[clowast 1] f(clowast2) is a subgradient on [0 clowast] and the average density on [clowast2 c] equals f(clowast2)because F (clowast) intersects the subgradient

reduces to F being concave on [clowast 1] Linear loss utility is also helpful to interpret the otherconditions Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom clowast2 to clowast be simultaneously less than that from clowast2 to t for all t isin (clowast2 clowast] and greaterthan that from s to clowast2 for all s isin [0 clowast2) Equivalently the average density from clowast2 to clowast

equals f(clowast2) and f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] The subgradientcondition is analogous to that discussed after Proposition 2 (More generally conditions (ii)and (iii) with clowast = 1 imply the condition of Proposition 2) The additional requirement ensuresthat the threshold clowast is an optimal threshold See Figure 1

Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper In fact the twoconditions are identical even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning15 Conditions (ii) and (iii) of Proposition 3 donrsquothave analogs in Amador and Bagwellrsquos work however because these concern optimalityconditions that turn on our status quo

Remark 3 For linear loss utility it can be verified that if F is strictly unimodal (ie strictlyconvex and then strictly concave) then interval delegation is optimal Either there will be aclowast isin [0 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will

15 Indeed we conjecture that our methodology elaborated in Subsection 34 can be used to show that Amadorand Bagwellrsquos (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms

14

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

be met and no compromise is optimal16 ⋄

We can extend this observation as follows

Corollary 3 Assume Condition LQ Interval delegation is optimal if the type density f is logconcaveon [0 1] if in addition f is strictly logconcave on [0 1] or Proposerrsquos utility is strictly concave thenthere is a unique optimal interval

Recall that logconcavity is stronger than unimodality but many familiar distributions havelogconcave densities including the uniform normal and exponential distributions (Bagnoliand Bergstrom 2005) The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 1] the set of optimal interval thresholds is connectedif [clowast1 1] and [clowast2 1] are both optimal interval delegation sets then so is [clowast 1] for all clowast isin [clowast1 c

lowast2]

Such multiplicity arises under the uniform distribution and linear loss utility Either strict log-concavity of the type density or strict concavity of Proposerrsquos utility eliminates multiplicity

34 Methodology

Let us outline the idea behind the proofs of Propositions 1ndash3 We use a Lagrangian methodas has proved fruitful in prior work on optimal delegation notably in Amador and Bagwell(2013) However the presence of a status quo requires some differences in our approachIn particular while prior work has largely focussed on optimality of connected delegationsets our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo Moreover our approach provides asimple way to incorporate stochastic mechanisms which as already highlighted are oftennot addressed in prior work

Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms

maxαisinA

983133 983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds ge 0 forallv isin [v v]

16 Let Mo be the (strictly positive for simplicity) mode of F and let ∆(x) = F (2x)minusF (x)x minus f(x) F being

convex-concave implies that letting clowast2 = maxx gt 0 ∆(x) = 0 ∆(x) ge 0 for x isin (0 clowast2) and ∆(x) le 0 forx isin (clowast2 1] Clearly clowast2 le Mo le clowast and hence f is decreasing on [clowast 1] The convex-concave property impliesthat f(clowast2) is a subgradient of F at clowast2 on the domain [0 clowast] and if clowast2 gt 12 then f(12) is a subgradient ofF on the domain [0 1]

15

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

Problem (R) is well-behaved because the constraint set is convex and owing to κ equivinfaisin[01) minusuprimeprime(a) the objective is a concave functional of α It differs from (D) in two waysFirst the constraint has been relaxed IC requires the inequality to hold with equality Secondthe objective has been modified to incorporate a penalty for violating IC Plainly if αlowast is ICand a solution to problem (R) then it is also a solution to (D) But we establish (see Lemma A1in Appendix A) that in this case αlowast is also a solution to problem (P) ie it is optimal amongstochastic mechanisms The idea is as follows If there were a stochastic mechanism that isstrictly better than αlowast in problem (P) consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome While this mechanism would not be IC ingeneral we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value contradicting the optimality of αlowast in (R) Establishing a higher valuerelies on the objective in (R) being a concave functional

The sufficiency results in Propositions 1ndash3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R) It is here that our approachinvolves constructing suitable Lagrange multipliers

For the necessity results we first establish in Lemma A4 in Appendix A that under Con-dition LQ if a deterministic mechanism αlowast solves problem (P) then it also solves problem(R) The idea is as follows Suppose a solution α to problem (R) provides a strictly highervalue than αlowast We construct a corresponding IC mechanism m such that α(v) = Em(v)[a] for allv Roughly monotonicity of α (by definition of the set A) implies existence of transfers thatmake α IC in a quasi-linear model the inequality constraints in (R) mean the transfers can bechosen to be positive (ie they can be viewed as money burning) and because of Vetoerrsquosquadratic utility positive transfers can be substituted for by the action variance of suitablelotteries Since uprimeprime(a) = minusκ for a lt 1 under Condition LQ the condition makes the objectivein (R) a linear functional in the relevant domain We can thus show that mechanism m obtainsa strictly higher value than αlowast in (P) a contradiction

We then establish necessity of the conditions in Propositions 1ndash3 by showing that unlessthese conditions are satisfied the corresponding mechanisms can be strictly improved uponin problem (R) Here we use the fact that the constraint set in (R) is convex and thereforefirst-order conditions must hold at a solution More specifically the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative

16

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

4 Comparative Statics and Comparisons

41 Comparative Statics

We derive two comparative statics restricting attention to interval delegation This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3) or just because such menus are simple tractable or relevant for applications

If Proposer proposes A = [c 1] with c isin [0 1] then Vetoer chooses 0 if v lt c2 c if v isin[c2 c] v if v isin [c 1] and 1 if v gt 1 Hence Proposerrsquos expected utility or welfare fromA = [c 1] is

W (c) = u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv

Differentiating the first-order condition for clowast isin (0 1) to be an optimal threshold amonginterval delegation sets is that it must be a zero of

2uprime(clowast) [F (clowast)minus F (clowast2)]minus f(clowast2) [u(clowast)minus u(0)] (2)

In general there can be multiple optimal thresholds even among interior thresholds Ac-cordingly let the set of optimal thresholds for interval delegation be

Clowast = arg maxcisin[01]

W (c)

We use the strong set order to state comparative statics Recall that for X Y sube R Xis larger than Y in the strong set order denoted X geSSO Y if for any x isin X and y isin Y minx y isin Y and maxx y isin X We say that Clowast increases (resp decreases) if it gets larger(resp smaller) in the strong set order Since interval delegation with a lower threshold givesVetoer a superset of options to choose from a decrease in Clowast corresponds to offering morediscretion It can also be interpreted as Proposer compromising more As mentioned afterCorollary 3 under Condition LQ and a logconcave type density Clowast is a (closed) interval Inthat case a decrease in Clowast is equivalent to a decrease in both minClowast and maxClowast

Our comparative statics concern Proposerrsquos risk aversion and the ex-ante preference align-ment between Proposer and Vetoer We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant regionminusuprimeprime(a)uprime(a) strictly increases for all a isin [0 1) As is well known such a change can also beexpressed in terms of concave transformations of Proposerrsquos utility Under Condition LQ it

17

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

corresponds to a higher weight on the quadratic term We say that the two players are strictlymore aligned if Vetoerrsquos ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 1] for all 0 le vL lt vH le 1 f(vH)f(vL) lt g(vH)g(vL)

Proposition 4 Among interval delegation sets there is

(i) more discretion (ie Clowast decreases) if Proposer becomes strictly more risk averse and

(ii) less discretion (ie Clowast increases) if Vetoer becomes strictly more aligned with Proposer

The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty specifically Karlinrsquos (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994)

The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto and hence she compromises more The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto and henceshe compromises less Yet the precise conditions in the proposition are nuanced In partic-ular the stochastic ordering used in our notion of alignment is important one can constructexamples in which among interval delegation sets Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift) Furthermore absent the focus on interval delegation itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature The broad finding there is that among interval delegation greater preference similar-ity in a suitable sense leads to more discretion (Holmstrom 1984 Theorem 3) The differenceowes to and highlights the distinct rationales for discretion In those models the delegatorwould like to give the agent discretion to benefit from the agentrsquos expertise the degree ofdiscretion is limited by the extent of preference misalignment In our setting on the otherhand the agent is given discretion only because of her veto power greater ex-ante preferencealignment mitigates that concern

Example 1 Under Condition LQ the first-order condition for an optimal interval threshold(ie expression (2) equals zero) becomes

2(1 + γ minus 2γclowast) [F (clowast)minus F (clowast2)] = clowast(1 + γ minus γclowast)f(clowast2)

18

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

μ=15

μ=1

μ=05

μ=02

0 02 04 06γ0

02

04

06

08

1c

(a) σ = 1

γ=01

γ=02

γ=05

γ=099

0 02 04 06 08 1σ0

02

04

06

08

1c

(b) micro = 045

Figure 2 ndash Optimal interval thresholds for Normal distributions (mean micro variance σ2) andlinear-quadratic Proposer utility u(a) = minus(1minus γ)|1minus a|minus γ(1minus a)2

Recall from Corollary 3 that when combined with boundary conditions there will be aunique solution for any strictly logconcave type density moreover the corresponding intervalis then an (unrestricted) optimal mechanism Given uniqueness the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4 moreover the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions Figure 2 illustrates for Normal distributions The left panel verifiescomparative statics already discussed note that a higher mean micro is a likelihood ratio right-shift and hence more alignment17 The right panel shows comparative statics in the varianceof the distribution We see that there is less discretion when the variance is lower with theoptimal threshold converging as σ rarr 0 to Proposerrsquos optimal offer 09 to type micro = 045 ⋄

While we do not have general comparative statics results in the variability of the typedistribution it can be shown that for any strictly unimodal distribution with mode Mo ge 0an optimal interval delegation set has more compromise than if Proposer knew Vetoerrsquos typeto be Mo That is if clowast is an optimal interval threshold then clowast le min2Mo 1 the inequalityis strict if Mo isin (0 12) Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposerrsquos compromise

42 Comparisons

This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work

17 When micro ge 1 a higher micro can be viewed as shifting Vetoer overall further away to the right of Proposer butwhat is relevant is the change of the distribution on the interval [0 1]

19

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

A natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model Proposer makes a take-it-or-leave it proposal a isin R which Vetoer can accept orveto This can be viewed as restricting Proposer to singleton delegation sets Clearly Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism Weassume throughout this subsection that no compromise is not an optimal interval delegationset as noted in Subsection 32 it is sufficient that Proposerrsquos utility u(a) is differentiable at hisideal point a = 1 (hence uprime(1) = 0)18 We will see below that not only does Proposer strictlybenefit from optimal delegation but so does Vetoer under some conditions

Matthews (1989) studies cheap talk before veto bargaining prior to Proposer making asingleton proposal Vetoer can send a costless and nonbinding message As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium in which Proposermakes the same proposal aU gt 0 as he would absent the possibility of cheap talk Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk it is sufficient given the support of our type density that uprime(1) = 0 (or that eventhe weaker condition in fn 18 holds) A set of low Vetoer types pool on a ldquoveto threatrdquomessage while the complementary set of high types pool on an ldquoacquiescingrdquo message Inresponse to the latter message Proposer offers a = 1 in response to the veto threat Proposeroffers some aI isin (0 1) The former proposal is accepted by all types that acquiesced whilethe latter is accepted by only a subset of types that made the veto threat types below somestrictly positive threshold exercise the veto An influential cheap-talk equilibrium is outcomeequivalent to the delegation set aI 1 in our framework

There can be multiple cheap-talk equilibria with distinct outcomes both among influ-ential equilibria and among noninfluential equilibria (ie distinct aI and aU respectively)Matthews shows that aI lt aU in any two equilibria of the respective kinds moreover he pro-vides conditions under which aI is unique ie all influential cheap-talk equilibria have thesame outcome (Matthews 1989 Remark 3) As elaborated in the proof of our Proposition 5multiplicity is ruled out when the following function has a unique zero

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2)[u(a)minus u(0)] (3)

By way of comparison we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation setrsquos threshold

18 A weaker condition suffices 2uprime(1)[1minus F (12)] lt f(12)[u(1)minus u(0)] This ensures that 1 is not an optimalsingleton proposal nor is 1 an optimal interval delegation set Recall that when u is not differentiable at 1uprime(1) refers to the left derivative

20

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

Proposition 5 Assume no compromise is not an optimal delegation set and that either (2) or (3) isstrictly downcrossing on (0 1)19 Any optimal interval delegation set [clowast 1] has clowast lt minaI aU forany influential and noninfluential cheap-talk equilibrium aI and aU respectively Hence if [clowast 1] is anoptimal delegation set then it strongly Pareto dominates any cheap-talk outcome influential or not

By strong Pareto dominance we mean that Proposer is ex-ante better off while Vetoer isbetter off no matter his type moreover a set of Vetoer types that have strictly positive proba-bility are strictly better off Proposition 5rsquos conclusions hold trivially when full delegation isthe optimal delegation set (clowast = 0) even without its hypotheses But under its hypothesesthe conclusions also apply to other optimal intervals The function (2) is strictly downcross-ing on (0 1) under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposerrsquos utility Indeed this underlies the uniqueness claim in Corollary 3 seeLemma B1 Moreover Corollary 3 also assures that interval delegation is then optimal

Here is the intuition behind Proposition 5 Consider Proposerrsquos tradeoff when marginallylowering his proposal aI isin (0 1) in an influential cheap-talk equilibrium The benefit is thatsome types just below aI2 will accept rather than veto the cost is that the action inducedfrom all types in the interval (aI2 (1 + aI)2) is lower When Proposer instead delegatesthe interval [aI 1] the benefit from lowering aI is unchanged while the cost is largely obvi-ated as most types above aI2 are unaffected by the change Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk20

Consequently all Vetoer types benefitmdashat least weakly and some strictlymdashfrom optimal in-terval delegation as compared to cheap talk While Proposer could be harmed by a restrictionto interval delegation there is strong Pareto dominance when intervals constitute optimaldelegation

We note that if interval delegation is not optimal then some Vetoer types may be worse offunder optimal delegation than under cheap talk For example it is possible that the optimaldelegation set takes the form alowast 1 with alowast isin (0 1) In this case one can show that necessarilyalowast lt aI in any influential cheap-talk equilibrium intuitively while aI is sequentially rationalcommitting to a lower proposal helps ex ante by inducing action 1 rather than aI from sometypes Consequently while Proposer strictly benefits from optimal delegation some Vetoertypes would strictly prefer either cheap-talk outcome

19 A function h(a) is strictly downcrossing if for any aL lt aH h(aL) le 0 =rArr h(aH) lt 020 The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality

21

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

5 Applications

We now discuss some implications and interpretations of our analysis in the context ofthree applications

Menus of products Our framework can be applied to questions of which products to presentcustomers with albeit in a stylized manner For an illustration suppose a salesperson has athis disposal a set of products indexed by a isin [0 1] with higher a corresponding to higherquality The price of product a is ka2 where k gt 0 This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost Consumers vary in how they trade off qualityand price specifically a consumer of type v ge 0 has gross valuation va and hence net-of-pricepayoff vaminuska2 If a consumer does not purchase his payoff is 0 a consumer cannot purchasea product he is not shown (perhaps because of ignorance or because the salesperson canclaim it is unavailable) Observe that we can normalize k = 12 as this simply rescales theconsumer type v The salesperson receives a higher commission on better products reflectedby his strictly increasing and concave utility u(a) Given any belief density the salespersonholds about a particular consumerrsquos valuation the salespersonrsquos problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting

Take the case of a linear u Propositions 1ndash3 imply that if the density of v is logconcave it isoptimal to show the consumer some set of ldquobest productsrdquo (ie an interval of products [c 1])if the density is strictly decreasing then all products should be shown and if the density isstrictly increasing then only the highest-quality product should be shown Proposition 4(i)implies that if the commission schedule changes to make u more concave the salespersonshows a larger set of products Proposition 4(ii) implies that if wealthier consumers (or ifwealth is unobservable some proxy thereof) have a higher distribution of v in the likelihoodratio sense then wealthier consumers are shown a smaller set of products

What if a consumer can choose the information to disclose about her type21 Specificallysuppose as is standard in voluntary disclosure models that any type v can send any message(a closed subset of R+) that contains v The salesperson decides on the product menu afterobserving the message No matter the type distribution there are at least two equilibria onein which no type discloses any information and one in which all types fully disclose22 Every

21 Ali Lewis and Vasserman (2019) Hidir and Vellodi (2020) and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination

22 For any unused message V sube R+ let the salesperson put probability 1 on v = maxV (or if supV = infin onsome v isin V with v ge 1) and offer the correspondingly optimal singleton menu It is then straightforward that

22

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

consumer type prefers the former equilibrium to the latter some types have a strict preferenceunless nondisclosure results in only a single product being shown (Proposition 2) In generalthere can be other equilibria some of which may dominate the nondisclosure equilibriumin terms of ex-ante consumer welfare23 However when the salesperson offers all productsunder the prior (Proposition 1) the nondisclosure equilibrium is consumer optimalmdashnot onlyex ante but for every consumer type

Lesser-included offenses The legal doctrine of lesser-included offenses in criminal cases isldquothe concept that a defendant may be found guilty of an uncharged lesser offense insteadof the offenses formally charged a recognized and well-established feature of the Amer-ican criminal justice systemrdquo (Adlestein 1995) For instance ldquothe lesser-included offensesof first degree murder include second degree murder voluntary manslaughter involuntarymanslaughter criminally negligent homicide and aggravated assaultrdquo (Orzach and Spurr2008)

Our model views v as a juryrsquos (or judgersquos) evaluation of the optimal penalty or true sever-ity of a crime and assumes that the jury will convict the defendant of the closest chargeto v that is available Verdict 0 corresponds to a complete acquittal which is always avail-able to the jury while 1 is the maximum penalty A prosecutor can put forward any set ofcharges in [0 1] and he seeks to maximize the penalty24 Assume his utility u is concave Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a

is included then the jury can choose any verdict in [0 a] The prosecutor is ex-ante worseoff with such constraint often strictly so However our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor as the outcome is the same with orwithout the doctrine In fact following our discussion after Corollary 1 particularly the endof fn 12 when Proposition 1rsquos requirement holds the prosecutor is not hurt ex ante from thedoctrine being attached to any set of charges not only his optimum This applies when the

no consumer type does strictly better by deviating to any unused message23 For example suppose the type density is strictly decreasing on a small interval [0 δ] and strictly increas-

ing thereafter and u is linear Then under nondisclosure the salespersonrsquos optimal menu is the singleton 1(Proposition 2) There is also a partial-disclosure equilibrium in which types [0 δ] pool on the message [0 δ] andall higher types pool on the message [δinfin) the former message leads to the menu [0 δ] by the full-delegationlogic of Corollary 1 while the latter message leads to the singleton 1 by the no-compromise logic of Proposi-tion 2 Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium somestrictly

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971) Ourassumption on the juryrsquos behavior is compatible with it acting on the basis of (expected) utility but could alsobe viewed as a reduced form It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met

23

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

prosecutor is sufficiently risk averse (Corollary 2)

Conversely consider the expected utility of a defendant whose utility is strictly decreasingon [0 1] As this is the additive inverse of some prosecutor utility function the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine25

She is not assured an improvement ex post however For example when no compromiseis unconstrained optimal for the prosecutor (Proposition 2) the doctrine strictly harms thedefendant when v isin (0 12) More generally the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutorThe fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid26

We note that if the prosecutor were restricted to bringing a single charge then under thecondition of Proposition 1 the lesser-included offenses doctrine would benefit him and hurta defendant with utility minusu Moreover regardless of whether full delegation is unconstrainedoptimal for the prosecutor the doctrine would lead him to bringing the maximum chargewhereas absent the doctrine the optimal single charge would typically not be the maximum

Legislatures Executives and Bureaucracies Legislatures write bills that can be vetoed byexecutives But executives do more than just approve or veto as emphasized by Epstein andOrsquoHalloran (1996 pp 378ndash379) ldquoall laws passed by Congress are implemented by the execu-tive branch in one form or anotherrdquo and since ldquoPresidents generally appoint administratorswith preferences similar to their ownrdquo the amount of discretion given is a ldquokey variable in congressional-executive relationsrdquo One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature in the sense of Proposition 4(ii) This flips the com-parative static emphasized in the political science literature (Epstein and OrsquoHalloran 19961999) which stems from expertise-based delegation models27 Of course in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees

25 This observation holds regardless of the curvature of the defendantrsquos utility w(a) because a prosecutor withutility minusw(a) is ex-ante worse off with a constraint on his delegation set even if contrary to our maintainedassumption minusw is not concave

26 Eg in Shrum v State (1999) the defendant who was convicted for manslaughter on a charge of murderappealed that ldquo1) heat of passion manslaughter is not a lsquonecessarily included offensersquo of premeditated murder2) a jury must acquit when the evidence supports a charge not alleged in the Informationrdquo The appeal wasdenied

27 For exceptions and caveats see for example Volden (2002) and Huber and McCarty (2004) Volden (2002p 112) notes that modeling the executiversquos veto is important for his finding that ldquothere are conditions under

24

There are examples of legislatures apparently providing misaligned executives greater dis-cretion because of veto power A case in point is the ldquoUS Troop Readiness Veteransrsquo CareKatrina Recovery and Iraq Accountability Appropriations Actrdquo (House Resolution 2206) en-acted in May 2007 that provided funding for the United Statesrsquo war on Iraq Congress passedan earlier version House Resolution 1591 that set a deadline of April 2008 for US troops towithdraw from Iraq This suggests that even after accounting for any expertise-based del-egation rationale Congress preferred a relatively tight deadline But the bill was vetoed byPresident George W Bush Although Democrats controlled both chambers of Congress theydid not have the requisite supermajority to override the veto To secure the Presidentrsquos ap-proval the eventual Act replaced the withdrawal deadline with vague metrics that gave thePresident more discretion

We must also stress an alternative perspective on our results rather than passing bills thatgrant ex-post discretion discretion can manifest in the executive effectively selecting whichbill (from some subset none of which grant ex-post discretion) the legislature passes Forexample the President may be consulted by Congress about different versions of legislationThese two forms of discretion are equivalent within our model An empirical test of ourmodelrsquos predictions in the political arena would have to overcome this challenge and that ofthe coexistence of expertise- and veto-based delegation rationales

6 Conclusion

We have studied Proposerrsquos optimal mechanism absent transfers in a simple model ofveto bargaining Our main results identify sufficient and necessary conditions for the optimalmechanism to take the form of certain delegation sets including full delegation no compro-mise and more generally interval delegation While we have focussed on a quadratic lossfunction for Vetoer our analytical methodology can be applied to deduce optimality of thesedelegation sets for a broader class of Vetoer preferences Specifically the methods can be read-ily applied to Vetoer utility functions of the form va + b(a) for any differentiable and strictlyconcave function b The conditions in Propositions 1ndash3 would be more complicated howeverOur methodology could also be used to deduce optimality of other kinds of delegation setsfor example Proposer offering his ideal point and one additional compromise option

A key assumption underlying our analysis is that of Proposer commitment In some con-

which discretion is increased upon a divergence in legislative-executive preferencesrdquo The mechanism under-lying his findings is different from that in this paper however in particular expertise-based delegation is stillessential to his analysis

25

texts Proposer may be unable to preclude reapproaching Vetoer with another proposal (ormenu of proposals) following a veto When the optimal mechanism in our setting is fulldelegation we believe that such lack of commitment is not problematic By offering the full-delegation menu to begin with bargaining will effectively conclude at the first opportunity

When full delegation is not optimal however matters are considerable more nuanced Se-quential veto bargaining without commitment has received only limited theoretical attentionlargely in finite-horizon models with particular type distributions (eg Cameron 2000) Inongoing research we are studying an infinite-horizon model Our preliminary results sug-gest that owing to single-peaked preferences non-Coasian dynamics can emerge that allowProposer to obtain her commitment solution when players are patient

ReferencesADLESTEIN A L (1995) ldquoConflict of the Criminal Statute of Limitations with Lesser Offenses

at Trialrdquo Wm amp Mary L Rev 37 199

ALI S N G LEWIS AND S VASSERMAN (2019) ldquoVoluntary Disclosure and PersonalizedPricingrdquo Unpublished

ALONSO R AND N MATOUSCHEK (2008) ldquoOptimal Delegationrdquo Review of Economic Studies75 259ndash293

AMADOR M AND K BAGWELL (2013) ldquoThe Theory of Optimal Delegation with an Appli-cation to Tariff Capsrdquo Econometrica 81 1541ndash1599

mdashmdashmdash (2019) ldquoRegulating a Monopolist With Uncertain Costs Without Transfersrdquo Unpub-lished

AMADOR M K BAGWELL AND A FRANKEL (2018) ldquoA Note on Interval DelegationrdquoEconomic Theory Bulletin 6 239ndash249

AMADOR M I WERNING AND G M ANGELETOS (2006) ldquoCommitment vs FlexibilityrdquoEconometrica 74 365ndash396

AMBRUS A AND G EGOROV (2017) ldquoDelegation and Nonmonetary Incentivesrdquo Journal ofEconomic Theory 171 101ndash135

ARROW K J (1965) ldquoAspects of the Theory of Risk-Bearingrdquo Helsinki Yrjo Jahnssoninsaatio lecture 2

26

BAGNOLI M AND T BERGSTROM (2005) ldquoLog-concave Probability and its ApplicationsrdquoEconomic Theory 26 445ndash469

BARON D P AND R B MYERSON (1982) ldquoRegulating a Monopolist with Unknown CostsrdquoEconometrica 50 911ndash930

CAMERON C AND N MCCARTY (2004) ldquoModels of Vetoes and Veto Bargainingrdquo AnnualReview of Political Science 7 409ndash435

CAMERON C M (2000) Veto Bargaining Presidents and the Politics of Negative Power Cam-bridge University Press

COMPTE O AND P JEHIEL (2009) ldquoVeto Constraint in Mechanism Design Inefficiency withCorrelated Typesrdquo American Economic Journal Microeconomics 1 182ndash206

DWORCZAK P AND G MARTINI (2019) ldquoThe Simple Economics of Optimal PersuasionrdquoJournal of Political Economy 127 1993ndash2048

EPSTEIN D AND S OrsquoHALLORAN (1996) ldquoDivided Government and the Design of Admin-istrative Procedures A Formal Model and Empirical Testrdquo Journal of Politics 58 373ndash397

mdashmdashmdash (1999) Delegating Powers A Transaction Cost Politics Approach to Policy Making underSeparate Powers Cambridge University Press

FORGES F AND J RENAULT (2020) ldquoStrategic Information Transmission with Senderrsquos Ap-provalrdquo Unpublished

HIDIR S AND N VELLODI (2020) ldquoPersonalization Discrimination and Information Disclo-surerdquo Forthcoming in Journal of the European Economic Association

HOLMSTROM B (1977) ldquoOn Incentives and Control in Organizationsrdquo PhD Thesis Stan-ford University

mdashmdashmdash (1984) ldquoOn the Theory of Delegationrdquo in Bayesian Models in Economic Theory ed byM Boyer and R Kihlstrom North Holland 115ndash141

HUBER J D AND N MCCARTY (2004) ldquoBureaucratic Capacity Delegation and PoliticalReformrdquo American Political Science Review 98 481ndash494

ICHIHASHI S (2020) ldquoOnline Privacy and Information Disclosure by Consumersrdquo AmericanEconomic Review 110 569ndash95

27

KARLIN S (1968) Total Positivity vol I Stanford California Stanford University Press

KOLOTILIN A (2018) ldquoOptimal Information Disclosure A Linear Programming ApproachrdquoTheoretical Economics 13 607ndash635

KOLOTILIN A AND A ZAPECHELNYUK (2019) ldquoPersuasion meets Delegationrdquo Unpub-lished

KOVAC E AND T MYLOVANOV (2009) ldquoStochastic Mechanisms in Settings Without Mone-tary Transfers The Regular Caserdquo Journal of Economic Theory 144 1373ndash1395

LANDES W M (1971) ldquoAn Economic Analysis of the Courtsrdquo Journal of Law and Economics14 61ndash107

MATTHEWS S A (1989) ldquoVeto Threats Rhetoric in a Bargaining Gamerdquo Quarterly Journal ofEconomics 104 347ndash369

MELUMAD N AND T SHIBANO (1991) ldquoCommunication in Settings with No TransfersrdquoRAND Journal of Economics 22 173ndash198

MILGROM P AND I SEGAL (2002) ldquoEnvelope Theorems for Arbitrary Choice Setsrdquo Econo-metrica 70 583ndash601

MILGROM P AND C SHANNON (1994) ldquoMonotone Comparative Staticsrdquo Econometrica 157ndash180

ORZACH R AND S J SPURR (2008) ldquoLesser-included Offensesrdquo International Review of Lawand Economics 28 239ndash245

PRATT J W (1964) ldquoRisk Aversion in the Small and in the Largerdquo Econometrica 32 122ndash136

ROMER T AND H ROSENTHAL (1978) ldquoPolitical Resource Allocation Controlled Agendasand the Status Quordquo Public Choice 33 27ndash43

mdashmdashmdash (1979) ldquoBureaucrats Versus Voters On the Political Economy of Resource Allocationby Direct Democracyrdquo Quarterly Journal of Economics 93 563ndash587

ROYDEN H AND P FITZPATRICK (2010) Real Analysis Prentice-Hall Inc 4th ed

VOLDEN C (2002) ldquoA Formal Model of the Politics of Delegation in a Separation of PowersSystemrdquo American Journal of Political Science 46 111ndash133

ZAPECHELNYUK A (2019) ldquoOptimal Quality Certificationrdquo American Economic Review In-sights 2 161ndash176

28

Appendices

A Proofs of Propositions 1 2 and 3

In Appendix A we assume the support of the type distribution F is [0 1] This is withoutloss (even among stochastic mechanisms) because it is always optimal for Proposer to chooseaction 1 for types above 1 and given the outside option to choose action 0 for types below 0

A1 Sufficient Conditions

For convenience we recall Proposerrsquos problem (P)

maxmisinS

983133 1

0

Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046minus

983133 v

0

Em(x)[a]dx = 0 forallv isin [0 1] (IC-env)

We also recall the relaxed problem (R)

maxαisinA

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx ge 0 forallv isin [0 1]

We show first that any IC solution to the relaxed problem also solves problem (P)

Lemma A1 Suppose αlowast isin A solves problem (R) and is incentive compatible Then αlowast also solves(P)

Proof To obtain a contradiction let m isin S be feasible for (P) and suppose it achieves a strictlyhigher objective value in (P) than αlowast Define α isin A by setting α(v) = Em(v)[a] for each vThen Jensenrsquos inequality implies Em(v) [av minus a22] ge vα(v)minusα(v)22 whereas

983125 v

0Em(x)[a]dx =

983125 v

0α(x)ds Hence feasibility of m in (P) implies feasibility of α in (R) Moreover

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v)

ge983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(x)[a]dx

983062dF (v)

=

983133 1

0

Em(v)[u(a)]dF (v)

29

gt

983133 1

0

u(αlowast(v))dF (v)

=

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(x)dx

983064983062dF (v)

where the first inequality holds because the first line is a concave functional (by the defini-tion of κ equiv infaisin[01) minusuprimeprime(a)) the first equality holds because m is feasible in (P) the secondinequality holds because of our assumption that m achieves a strictly higher value than αlowastand the final equality holds because αlowast being IC implies it is feasible in (P) Therefore αlowast isnot optimal in (R) a contradiction

To show that a given delegation set solves the relaxed problem we construct Lagrangemultipliers and show that the induced action rule maximizes the following Lagrangean Givenα isin A and an increasing and right-continuous function Λ(v) let the Lagrangean be given by

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus κf(v)

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983062dΛ(v)

=

983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v) +

983133 1

0

α(v)dv[κF (1)minus Λ(1)] (A1)

where the second equality follows from integration by parts28

Lemma A2 Let αlowast be induced by a delegation set29 Suppose there is an increasing and right-continuous function Λ such that L(αlowastΛ) ge L(αΛ) for all α isin A Then α solves problem (R)

Proof Under the conditions stated in the lemma Theorem 1 in Amador and Bagwell (2013)implies that αlowast solves the relaxed problem To see this let their f be the negative of theobjective function in (R) X and Z be the vector space of bounded measurable functions andΩ = A let P = z isin Z|forallv isin [0 1] z(v) ge 0 and G be the negative of the left-hand side of theconstraint in (R) Define the linear function T Z rarr R by T (z) =

983125 1

0z(v)dΛ(v) (which is well-

defined since Λ corresponds to a finite measure (Royden and Fitzpatrick 2010 p 437) and

28983125 v

0α(s)ds is continuous and κF (v) minus Λ(v) has bounded variation as the difference of two increasing func-

tions Hence the Riemann-Stieltjes integral983125 1

0

983125 v

0α(s)dsd[κF (v)minus Λ(v)] exists and integration by parts is valid

29 That is there is some delegation set A such that αlowast(v) is an action in A cup 0 that type v prefers the most

30

the integral therefore exists because z is bounded and measurable (Royden and Fitzpatrick2010 p 375)) and observe that for all z isin P T (z) ge 0 and hence condition (i) holds Sinceαlowast is incentive compatible minusG(αlowast) isin P (condition (iii) holds) and T (G(αlowast)) = 0 (condition(iv) holds) Since αlowast maximizes the Lagrangean by assumption condition (ii) holds and weconclude that αlowast solves problem (R)

The follow observation will allow us to establish that the Lagrangean is a concave func-tional of α if Λ is increasing

Lemma A3 Suppose K is right-continuous and increasing and h R2 rarr R is bounded measurableand for each value of its second argument concave in its first argument Then the function S Linfin rarr Rdefined by S(α) =

983125 1

0h(α(v) v)dK(v) is concave

Proof Fix α1α2 isin Linfin c isin (0 1) and let αc = cα1 + (1minus c)α2 Then

S(αc)minus cS(α1)minus (1minus c)S(α2) =

983133 1

0

983043h(αc(v) v)minus ch(α1(v) v)minus (1minus c)h(α2(v) v)

983044dK(v) ge 0

because concavity of h implies that the integrand is positive for each v and because K isincreasing

Note that for each v u(α(v))f(v)+ κf(v)α(v)2

2is concave in α(v) since its second derivative

is given by f(v)[uprimeprime(α(v)) + κ] which is negative by definition of κ This implies that for eachv each integrand in (A1) is a concave function of α(v) Since Λ is increasing Lemma A3implies that the Lagrangean is concave in α

This implies that the Lagrangean is maximized at α if the Gateaux differential satisfiespartL(ααminus αΛ) le 0 for all α isin A

If Λ(1) = κF (1) (A1) simplifies to

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v)

The Gateaux differential is

partL(ααΛ) =983133 1

0

983059983147uprime(α(v))f(v)minus κF (v) + Λ(v)

983148α(v)

983060dv +

983133 1

0([v minus α(v)]α(v)) d[Λ(v)minus κF (v)]

(A2)

31

=

983133 1

0

983061983133 1

vuprime(α(x))f(x)minus κF (x) + Λ(x)dx

983062dα(v) +

983133 1

0

983061983133 1

v[xminus α(x)]d[Λ(x)minus κF (x)]

983062dα(v)

(A3)

where the second equality obtains using integration by parts

Below we construct increasing and right-continuous Lagrange multipliers that satisfyΛ(1) = κF (1) such that partL(ααminus αΛ) le 0

We begin with the optimality of full delegation

Proof of sufficiency part of Proposition 1 Note that the action function induced by full del-egation is α(v) = v We claim that α maximizes the Lagrangean for the multiplier Λ(v) =

κF (v) minus uprime(v)f(v) for v lt 1 and Λ(1) = κF (1) Note that the multiplier is increasing sinceκF (v) minus uprime(v)f(v) is increasing by assumption and uprime(v) ge 0 The Lagrangean is thereforemaximized at α if partL(ααminusαΛ) le 0 for all α isin A Note that the integrand of the first integralin (A2) is 0 for almost every v by choice of Λ and the second integral is 0 since α(v) = v

We next consider the optimality of no compromise

Proof of sufficiency part of Proposition 2 Note that the action rule induced by no compro-mise satisfies α(v) = 0 for v isin [0 1

2) and α(v) = 1 for v isin [1

2 1] Now suppose for all s isin [0 12)

and t isin (12 1] we have

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus s

and let ψ = inftisin(121](uprime(1) + κ(1minus t))F (t)minusF (12)

tminus12 Define Λ(v) = κF (12)minusψ for v isin [0 1) and

Λ(1) = κF (1)

Let s isin (12 1] Note that integration by parts implies983125 12

svdF (v) = 12F (12)minus sF (s)minus

983125 12

sF (v)dv Since Λ(v) is constant on [0 1) the definition of ψ implies that for any s isin [0 12)

983133 12

s

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 12

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (12)minus F (s)] + 12[Λ(12)minus κF (12)]minus s[Λ(s)minus κF (s)]

=[uprime(0) + κs][F (12)minus F (s)]minus (12minus s)ψ le 0 (A4)

32

Similarly for any t isin (12 1]

983133 t

12

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 t

12

[v minus α(v)]d[Λ(v)minus κF (v)]

=uprime(1)[F (t)minus F (12)] + (tminus 1)[Λ(t)minus κF (t)] + 12[Λ(12)minus κF (12)]

=[uprime(1) + κ(1minus t)][F (t)minus F (12)]minus (tminus 12)ψ ge 0 (A5)

Fix arbitrary α isin A that satisfies α(1) = 1 It follows from (A3) and the definition of α that

partL(ααminus αΛ)

=

983133 1

0

983063983133 1

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 1

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983064d[α(v)minus α(v)]

=

983133 1

0

983077983133 12

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 12

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983078dα(v)

Since α is increasing (A4) and (A5) imply that partL(αα minus αΛ) le 0 Since the optimal actionrule chooses action 1 for type 1 we conclude that α is optimal

Lastly we consider optimality of interval delegation

Proof of sufficiency part of Proposition 3 The induced action function is α(v) = 0 for v lt

clowast2 α(v) = clowast for clowast2 le v le clowast and α(v) = v for v gt clowast

We propose the following multiplier

Λ(v) =

983099983105983105983105983103

983105983105983105983101

κF (clowast2)minus uprime(clowast)F (clowast)minusF (clowast2)clowastminusclowast2 if v lt clowast

κF (v)minus uprime(v)f(v) if clowast le v lt 1

κF (1) if v = 1

Λ is constant on [0 clowast) and it follows from Proposition 3rsquos condition (i) that Λ is increasing on(clowast 1] To see that Λ is increasing at clowast note that condition (ii) holds as an equality for t = clowast

and hence the derivative of the LHS of condition (ii) with respect to t must be negative att = clowast which yields

minusκF (clowast)minus F (clowast2)

clowast2+ uprime(clowast)

f(clowast)clowast2minus (F (clowast)minus F (clowast2))

(clowast2)2le 0

Hence Λ is increasing at clowast It is thus sufficient to show partL(ααminus αΛ) le 0 for all α isin A

33

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

texts Proposer may be unable to preclude reapproaching Vetoer with another proposal (ormenu of proposals) following a veto When the optimal mechanism in our setting is fulldelegation we believe that such lack of commitment is not problematic By offering the full-delegation menu to begin with bargaining will effectively conclude at the first opportunity

When full delegation is not optimal however matters are considerable more nuanced Se-quential veto bargaining without commitment has received only limited theoretical attentionlargely in finite-horizon models with particular type distributions (eg Cameron 2000) Inongoing research we are studying an infinite-horizon model Our preliminary results sug-gest that owing to single-peaked preferences non-Coasian dynamics can emerge that allowProposer to obtain her commitment solution when players are patient

ReferencesADLESTEIN A L (1995) ldquoConflict of the Criminal Statute of Limitations with Lesser Offenses

at Trialrdquo Wm amp Mary L Rev 37 199

ALI S N G LEWIS AND S VASSERMAN (2019) ldquoVoluntary Disclosure and PersonalizedPricingrdquo Unpublished

ALONSO R AND N MATOUSCHEK (2008) ldquoOptimal Delegationrdquo Review of Economic Studies75 259ndash293

AMADOR M AND K BAGWELL (2013) ldquoThe Theory of Optimal Delegation with an Appli-cation to Tariff Capsrdquo Econometrica 81 1541ndash1599

mdashmdashmdash (2019) ldquoRegulating a Monopolist With Uncertain Costs Without Transfersrdquo Unpub-lished

AMADOR M K BAGWELL AND A FRANKEL (2018) ldquoA Note on Interval DelegationrdquoEconomic Theory Bulletin 6 239ndash249

AMADOR M I WERNING AND G M ANGELETOS (2006) ldquoCommitment vs FlexibilityrdquoEconometrica 74 365ndash396

AMBRUS A AND G EGOROV (2017) ldquoDelegation and Nonmonetary Incentivesrdquo Journal ofEconomic Theory 171 101ndash135

ARROW K J (1965) ldquoAspects of the Theory of Risk-Bearingrdquo Helsinki Yrjo Jahnssoninsaatio lecture 2

26

BAGNOLI M AND T BERGSTROM (2005) ldquoLog-concave Probability and its ApplicationsrdquoEconomic Theory 26 445ndash469

BARON D P AND R B MYERSON (1982) ldquoRegulating a Monopolist with Unknown CostsrdquoEconometrica 50 911ndash930

CAMERON C AND N MCCARTY (2004) ldquoModels of Vetoes and Veto Bargainingrdquo AnnualReview of Political Science 7 409ndash435

CAMERON C M (2000) Veto Bargaining Presidents and the Politics of Negative Power Cam-bridge University Press

COMPTE O AND P JEHIEL (2009) ldquoVeto Constraint in Mechanism Design Inefficiency withCorrelated Typesrdquo American Economic Journal Microeconomics 1 182ndash206

DWORCZAK P AND G MARTINI (2019) ldquoThe Simple Economics of Optimal PersuasionrdquoJournal of Political Economy 127 1993ndash2048

EPSTEIN D AND S OrsquoHALLORAN (1996) ldquoDivided Government and the Design of Admin-istrative Procedures A Formal Model and Empirical Testrdquo Journal of Politics 58 373ndash397

mdashmdashmdash (1999) Delegating Powers A Transaction Cost Politics Approach to Policy Making underSeparate Powers Cambridge University Press

FORGES F AND J RENAULT (2020) ldquoStrategic Information Transmission with Senderrsquos Ap-provalrdquo Unpublished

HIDIR S AND N VELLODI (2020) ldquoPersonalization Discrimination and Information Disclo-surerdquo Forthcoming in Journal of the European Economic Association

HOLMSTROM B (1977) ldquoOn Incentives and Control in Organizationsrdquo PhD Thesis Stan-ford University

mdashmdashmdash (1984) ldquoOn the Theory of Delegationrdquo in Bayesian Models in Economic Theory ed byM Boyer and R Kihlstrom North Holland 115ndash141

HUBER J D AND N MCCARTY (2004) ldquoBureaucratic Capacity Delegation and PoliticalReformrdquo American Political Science Review 98 481ndash494

ICHIHASHI S (2020) ldquoOnline Privacy and Information Disclosure by Consumersrdquo AmericanEconomic Review 110 569ndash95

27

KARLIN S (1968) Total Positivity vol I Stanford California Stanford University Press

KOLOTILIN A (2018) ldquoOptimal Information Disclosure A Linear Programming ApproachrdquoTheoretical Economics 13 607ndash635

KOLOTILIN A AND A ZAPECHELNYUK (2019) ldquoPersuasion meets Delegationrdquo Unpub-lished

KOVAC E AND T MYLOVANOV (2009) ldquoStochastic Mechanisms in Settings Without Mone-tary Transfers The Regular Caserdquo Journal of Economic Theory 144 1373ndash1395

LANDES W M (1971) ldquoAn Economic Analysis of the Courtsrdquo Journal of Law and Economics14 61ndash107

MATTHEWS S A (1989) ldquoVeto Threats Rhetoric in a Bargaining Gamerdquo Quarterly Journal ofEconomics 104 347ndash369

MELUMAD N AND T SHIBANO (1991) ldquoCommunication in Settings with No TransfersrdquoRAND Journal of Economics 22 173ndash198

MILGROM P AND I SEGAL (2002) ldquoEnvelope Theorems for Arbitrary Choice Setsrdquo Econo-metrica 70 583ndash601

MILGROM P AND C SHANNON (1994) ldquoMonotone Comparative Staticsrdquo Econometrica 157ndash180

ORZACH R AND S J SPURR (2008) ldquoLesser-included Offensesrdquo International Review of Lawand Economics 28 239ndash245

PRATT J W (1964) ldquoRisk Aversion in the Small and in the Largerdquo Econometrica 32 122ndash136

ROMER T AND H ROSENTHAL (1978) ldquoPolitical Resource Allocation Controlled Agendasand the Status Quordquo Public Choice 33 27ndash43

mdashmdashmdash (1979) ldquoBureaucrats Versus Voters On the Political Economy of Resource Allocationby Direct Democracyrdquo Quarterly Journal of Economics 93 563ndash587

ROYDEN H AND P FITZPATRICK (2010) Real Analysis Prentice-Hall Inc 4th ed

VOLDEN C (2002) ldquoA Formal Model of the Politics of Delegation in a Separation of PowersSystemrdquo American Journal of Political Science 46 111ndash133

ZAPECHELNYUK A (2019) ldquoOptimal Quality Certificationrdquo American Economic Review In-sights 2 161ndash176

28

Appendices

A Proofs of Propositions 1 2 and 3

In Appendix A we assume the support of the type distribution F is [0 1] This is withoutloss (even among stochastic mechanisms) because it is always optimal for Proposer to chooseaction 1 for types above 1 and given the outside option to choose action 0 for types below 0

A1 Sufficient Conditions

For convenience we recall Proposerrsquos problem (P)

maxmisinS

983133 1

0

Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046minus

983133 v

0

Em(x)[a]dx = 0 forallv isin [0 1] (IC-env)

We also recall the relaxed problem (R)

maxαisinA

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx ge 0 forallv isin [0 1]

We show first that any IC solution to the relaxed problem also solves problem (P)

Lemma A1 Suppose αlowast isin A solves problem (R) and is incentive compatible Then αlowast also solves(P)

Proof To obtain a contradiction let m isin S be feasible for (P) and suppose it achieves a strictlyhigher objective value in (P) than αlowast Define α isin A by setting α(v) = Em(v)[a] for each vThen Jensenrsquos inequality implies Em(v) [av minus a22] ge vα(v)minusα(v)22 whereas

983125 v

0Em(x)[a]dx =

983125 v

0α(x)ds Hence feasibility of m in (P) implies feasibility of α in (R) Moreover

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v)

ge983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(x)[a]dx

983062dF (v)

=

983133 1

0

Em(v)[u(a)]dF (v)

29

gt

983133 1

0

u(αlowast(v))dF (v)

=

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(x)dx

983064983062dF (v)

where the first inequality holds because the first line is a concave functional (by the defini-tion of κ equiv infaisin[01) minusuprimeprime(a)) the first equality holds because m is feasible in (P) the secondinequality holds because of our assumption that m achieves a strictly higher value than αlowastand the final equality holds because αlowast being IC implies it is feasible in (P) Therefore αlowast isnot optimal in (R) a contradiction

To show that a given delegation set solves the relaxed problem we construct Lagrangemultipliers and show that the induced action rule maximizes the following Lagrangean Givenα isin A and an increasing and right-continuous function Λ(v) let the Lagrangean be given by

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus κf(v)

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983062dΛ(v)

=

983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v) +

983133 1

0

α(v)dv[κF (1)minus Λ(1)] (A1)

where the second equality follows from integration by parts28

Lemma A2 Let αlowast be induced by a delegation set29 Suppose there is an increasing and right-continuous function Λ such that L(αlowastΛ) ge L(αΛ) for all α isin A Then α solves problem (R)

Proof Under the conditions stated in the lemma Theorem 1 in Amador and Bagwell (2013)implies that αlowast solves the relaxed problem To see this let their f be the negative of theobjective function in (R) X and Z be the vector space of bounded measurable functions andΩ = A let P = z isin Z|forallv isin [0 1] z(v) ge 0 and G be the negative of the left-hand side of theconstraint in (R) Define the linear function T Z rarr R by T (z) =

983125 1

0z(v)dΛ(v) (which is well-

defined since Λ corresponds to a finite measure (Royden and Fitzpatrick 2010 p 437) and

28983125 v

0α(s)ds is continuous and κF (v) minus Λ(v) has bounded variation as the difference of two increasing func-

tions Hence the Riemann-Stieltjes integral983125 1

0

983125 v

0α(s)dsd[κF (v)minus Λ(v)] exists and integration by parts is valid

29 That is there is some delegation set A such that αlowast(v) is an action in A cup 0 that type v prefers the most

30

the integral therefore exists because z is bounded and measurable (Royden and Fitzpatrick2010 p 375)) and observe that for all z isin P T (z) ge 0 and hence condition (i) holds Sinceαlowast is incentive compatible minusG(αlowast) isin P (condition (iii) holds) and T (G(αlowast)) = 0 (condition(iv) holds) Since αlowast maximizes the Lagrangean by assumption condition (ii) holds and weconclude that αlowast solves problem (R)

The follow observation will allow us to establish that the Lagrangean is a concave func-tional of α if Λ is increasing

Lemma A3 Suppose K is right-continuous and increasing and h R2 rarr R is bounded measurableand for each value of its second argument concave in its first argument Then the function S Linfin rarr Rdefined by S(α) =

983125 1

0h(α(v) v)dK(v) is concave

Proof Fix α1α2 isin Linfin c isin (0 1) and let αc = cα1 + (1minus c)α2 Then

S(αc)minus cS(α1)minus (1minus c)S(α2) =

983133 1

0

983043h(αc(v) v)minus ch(α1(v) v)minus (1minus c)h(α2(v) v)

983044dK(v) ge 0

because concavity of h implies that the integrand is positive for each v and because K isincreasing

Note that for each v u(α(v))f(v)+ κf(v)α(v)2

2is concave in α(v) since its second derivative

is given by f(v)[uprimeprime(α(v)) + κ] which is negative by definition of κ This implies that for eachv each integrand in (A1) is a concave function of α(v) Since Λ is increasing Lemma A3implies that the Lagrangean is concave in α

This implies that the Lagrangean is maximized at α if the Gateaux differential satisfiespartL(ααminus αΛ) le 0 for all α isin A

If Λ(1) = κF (1) (A1) simplifies to

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v)

The Gateaux differential is

partL(ααΛ) =983133 1

0

983059983147uprime(α(v))f(v)minus κF (v) + Λ(v)

983148α(v)

983060dv +

983133 1

0([v minus α(v)]α(v)) d[Λ(v)minus κF (v)]

(A2)

31

=

983133 1

0

983061983133 1

vuprime(α(x))f(x)minus κF (x) + Λ(x)dx

983062dα(v) +

983133 1

0

983061983133 1

v[xminus α(x)]d[Λ(x)minus κF (x)]

983062dα(v)

(A3)

where the second equality obtains using integration by parts

Below we construct increasing and right-continuous Lagrange multipliers that satisfyΛ(1) = κF (1) such that partL(ααminus αΛ) le 0

We begin with the optimality of full delegation

Proof of sufficiency part of Proposition 1 Note that the action function induced by full del-egation is α(v) = v We claim that α maximizes the Lagrangean for the multiplier Λ(v) =

κF (v) minus uprime(v)f(v) for v lt 1 and Λ(1) = κF (1) Note that the multiplier is increasing sinceκF (v) minus uprime(v)f(v) is increasing by assumption and uprime(v) ge 0 The Lagrangean is thereforemaximized at α if partL(ααminusαΛ) le 0 for all α isin A Note that the integrand of the first integralin (A2) is 0 for almost every v by choice of Λ and the second integral is 0 since α(v) = v

We next consider the optimality of no compromise

Proof of sufficiency part of Proposition 2 Note that the action rule induced by no compro-mise satisfies α(v) = 0 for v isin [0 1

2) and α(v) = 1 for v isin [1

2 1] Now suppose for all s isin [0 12)

and t isin (12 1] we have

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus s

and let ψ = inftisin(121](uprime(1) + κ(1minus t))F (t)minusF (12)

tminus12 Define Λ(v) = κF (12)minusψ for v isin [0 1) and

Λ(1) = κF (1)

Let s isin (12 1] Note that integration by parts implies983125 12

svdF (v) = 12F (12)minus sF (s)minus

983125 12

sF (v)dv Since Λ(v) is constant on [0 1) the definition of ψ implies that for any s isin [0 12)

983133 12

s

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 12

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (12)minus F (s)] + 12[Λ(12)minus κF (12)]minus s[Λ(s)minus κF (s)]

=[uprime(0) + κs][F (12)minus F (s)]minus (12minus s)ψ le 0 (A4)

32

Similarly for any t isin (12 1]

983133 t

12

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 t

12

[v minus α(v)]d[Λ(v)minus κF (v)]

=uprime(1)[F (t)minus F (12)] + (tminus 1)[Λ(t)minus κF (t)] + 12[Λ(12)minus κF (12)]

=[uprime(1) + κ(1minus t)][F (t)minus F (12)]minus (tminus 12)ψ ge 0 (A5)

Fix arbitrary α isin A that satisfies α(1) = 1 It follows from (A3) and the definition of α that

partL(ααminus αΛ)

=

983133 1

0

983063983133 1

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 1

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983064d[α(v)minus α(v)]

=

983133 1

0

983077983133 12

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 12

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983078dα(v)

Since α is increasing (A4) and (A5) imply that partL(αα minus αΛ) le 0 Since the optimal actionrule chooses action 1 for type 1 we conclude that α is optimal

Lastly we consider optimality of interval delegation

Proof of sufficiency part of Proposition 3 The induced action function is α(v) = 0 for v lt

clowast2 α(v) = clowast for clowast2 le v le clowast and α(v) = v for v gt clowast

We propose the following multiplier

Λ(v) =

983099983105983105983105983103

983105983105983105983101

κF (clowast2)minus uprime(clowast)F (clowast)minusF (clowast2)clowastminusclowast2 if v lt clowast

κF (v)minus uprime(v)f(v) if clowast le v lt 1

κF (1) if v = 1

Λ is constant on [0 clowast) and it follows from Proposition 3rsquos condition (i) that Λ is increasing on(clowast 1] To see that Λ is increasing at clowast note that condition (ii) holds as an equality for t = clowast

and hence the derivative of the LHS of condition (ii) with respect to t must be negative att = clowast which yields

minusκF (clowast)minus F (clowast2)

clowast2+ uprime(clowast)

f(clowast)clowast2minus (F (clowast)minus F (clowast2))

(clowast2)2le 0

Hence Λ is increasing at clowast It is thus sufficient to show partL(ααminus αΛ) le 0 for all α isin A

33

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

BAGNOLI M AND T BERGSTROM (2005) ldquoLog-concave Probability and its ApplicationsrdquoEconomic Theory 26 445ndash469

BARON D P AND R B MYERSON (1982) ldquoRegulating a Monopolist with Unknown CostsrdquoEconometrica 50 911ndash930

CAMERON C AND N MCCARTY (2004) ldquoModels of Vetoes and Veto Bargainingrdquo AnnualReview of Political Science 7 409ndash435

CAMERON C M (2000) Veto Bargaining Presidents and the Politics of Negative Power Cam-bridge University Press

COMPTE O AND P JEHIEL (2009) ldquoVeto Constraint in Mechanism Design Inefficiency withCorrelated Typesrdquo American Economic Journal Microeconomics 1 182ndash206

DWORCZAK P AND G MARTINI (2019) ldquoThe Simple Economics of Optimal PersuasionrdquoJournal of Political Economy 127 1993ndash2048

EPSTEIN D AND S OrsquoHALLORAN (1996) ldquoDivided Government and the Design of Admin-istrative Procedures A Formal Model and Empirical Testrdquo Journal of Politics 58 373ndash397

mdashmdashmdash (1999) Delegating Powers A Transaction Cost Politics Approach to Policy Making underSeparate Powers Cambridge University Press

FORGES F AND J RENAULT (2020) ldquoStrategic Information Transmission with Senderrsquos Ap-provalrdquo Unpublished

HIDIR S AND N VELLODI (2020) ldquoPersonalization Discrimination and Information Disclo-surerdquo Forthcoming in Journal of the European Economic Association

HOLMSTROM B (1977) ldquoOn Incentives and Control in Organizationsrdquo PhD Thesis Stan-ford University

mdashmdashmdash (1984) ldquoOn the Theory of Delegationrdquo in Bayesian Models in Economic Theory ed byM Boyer and R Kihlstrom North Holland 115ndash141

HUBER J D AND N MCCARTY (2004) ldquoBureaucratic Capacity Delegation and PoliticalReformrdquo American Political Science Review 98 481ndash494

ICHIHASHI S (2020) ldquoOnline Privacy and Information Disclosure by Consumersrdquo AmericanEconomic Review 110 569ndash95

27

KARLIN S (1968) Total Positivity vol I Stanford California Stanford University Press

KOLOTILIN A (2018) ldquoOptimal Information Disclosure A Linear Programming ApproachrdquoTheoretical Economics 13 607ndash635

KOLOTILIN A AND A ZAPECHELNYUK (2019) ldquoPersuasion meets Delegationrdquo Unpub-lished

KOVAC E AND T MYLOVANOV (2009) ldquoStochastic Mechanisms in Settings Without Mone-tary Transfers The Regular Caserdquo Journal of Economic Theory 144 1373ndash1395

LANDES W M (1971) ldquoAn Economic Analysis of the Courtsrdquo Journal of Law and Economics14 61ndash107

MATTHEWS S A (1989) ldquoVeto Threats Rhetoric in a Bargaining Gamerdquo Quarterly Journal ofEconomics 104 347ndash369

MELUMAD N AND T SHIBANO (1991) ldquoCommunication in Settings with No TransfersrdquoRAND Journal of Economics 22 173ndash198

MILGROM P AND I SEGAL (2002) ldquoEnvelope Theorems for Arbitrary Choice Setsrdquo Econo-metrica 70 583ndash601

MILGROM P AND C SHANNON (1994) ldquoMonotone Comparative Staticsrdquo Econometrica 157ndash180

ORZACH R AND S J SPURR (2008) ldquoLesser-included Offensesrdquo International Review of Lawand Economics 28 239ndash245

PRATT J W (1964) ldquoRisk Aversion in the Small and in the Largerdquo Econometrica 32 122ndash136

ROMER T AND H ROSENTHAL (1978) ldquoPolitical Resource Allocation Controlled Agendasand the Status Quordquo Public Choice 33 27ndash43

mdashmdashmdash (1979) ldquoBureaucrats Versus Voters On the Political Economy of Resource Allocationby Direct Democracyrdquo Quarterly Journal of Economics 93 563ndash587

ROYDEN H AND P FITZPATRICK (2010) Real Analysis Prentice-Hall Inc 4th ed

VOLDEN C (2002) ldquoA Formal Model of the Politics of Delegation in a Separation of PowersSystemrdquo American Journal of Political Science 46 111ndash133

ZAPECHELNYUK A (2019) ldquoOptimal Quality Certificationrdquo American Economic Review In-sights 2 161ndash176

28

Appendices

A Proofs of Propositions 1 2 and 3

In Appendix A we assume the support of the type distribution F is [0 1] This is withoutloss (even among stochastic mechanisms) because it is always optimal for Proposer to chooseaction 1 for types above 1 and given the outside option to choose action 0 for types below 0

A1 Sufficient Conditions

For convenience we recall Proposerrsquos problem (P)

maxmisinS

983133 1

0

Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046minus

983133 v

0

Em(x)[a]dx = 0 forallv isin [0 1] (IC-env)

We also recall the relaxed problem (R)

maxαisinA

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx ge 0 forallv isin [0 1]

We show first that any IC solution to the relaxed problem also solves problem (P)

Lemma A1 Suppose αlowast isin A solves problem (R) and is incentive compatible Then αlowast also solves(P)

Proof To obtain a contradiction let m isin S be feasible for (P) and suppose it achieves a strictlyhigher objective value in (P) than αlowast Define α isin A by setting α(v) = Em(v)[a] for each vThen Jensenrsquos inequality implies Em(v) [av minus a22] ge vα(v)minusα(v)22 whereas

983125 v

0Em(x)[a]dx =

983125 v

0α(x)ds Hence feasibility of m in (P) implies feasibility of α in (R) Moreover

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v)

ge983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(x)[a]dx

983062dF (v)

=

983133 1

0

Em(v)[u(a)]dF (v)

29

gt

983133 1

0

u(αlowast(v))dF (v)

=

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(x)dx

983064983062dF (v)

where the first inequality holds because the first line is a concave functional (by the defini-tion of κ equiv infaisin[01) minusuprimeprime(a)) the first equality holds because m is feasible in (P) the secondinequality holds because of our assumption that m achieves a strictly higher value than αlowastand the final equality holds because αlowast being IC implies it is feasible in (P) Therefore αlowast isnot optimal in (R) a contradiction

To show that a given delegation set solves the relaxed problem we construct Lagrangemultipliers and show that the induced action rule maximizes the following Lagrangean Givenα isin A and an increasing and right-continuous function Λ(v) let the Lagrangean be given by

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus κf(v)

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983062dΛ(v)

=

983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v) +

983133 1

0

α(v)dv[κF (1)minus Λ(1)] (A1)

where the second equality follows from integration by parts28

Lemma A2 Let αlowast be induced by a delegation set29 Suppose there is an increasing and right-continuous function Λ such that L(αlowastΛ) ge L(αΛ) for all α isin A Then α solves problem (R)

Proof Under the conditions stated in the lemma Theorem 1 in Amador and Bagwell (2013)implies that αlowast solves the relaxed problem To see this let their f be the negative of theobjective function in (R) X and Z be the vector space of bounded measurable functions andΩ = A let P = z isin Z|forallv isin [0 1] z(v) ge 0 and G be the negative of the left-hand side of theconstraint in (R) Define the linear function T Z rarr R by T (z) =

983125 1

0z(v)dΛ(v) (which is well-

defined since Λ corresponds to a finite measure (Royden and Fitzpatrick 2010 p 437) and

28983125 v

0α(s)ds is continuous and κF (v) minus Λ(v) has bounded variation as the difference of two increasing func-

tions Hence the Riemann-Stieltjes integral983125 1

0

983125 v

0α(s)dsd[κF (v)minus Λ(v)] exists and integration by parts is valid

29 That is there is some delegation set A such that αlowast(v) is an action in A cup 0 that type v prefers the most

30

the integral therefore exists because z is bounded and measurable (Royden and Fitzpatrick2010 p 375)) and observe that for all z isin P T (z) ge 0 and hence condition (i) holds Sinceαlowast is incentive compatible minusG(αlowast) isin P (condition (iii) holds) and T (G(αlowast)) = 0 (condition(iv) holds) Since αlowast maximizes the Lagrangean by assumption condition (ii) holds and weconclude that αlowast solves problem (R)

The follow observation will allow us to establish that the Lagrangean is a concave func-tional of α if Λ is increasing

Lemma A3 Suppose K is right-continuous and increasing and h R2 rarr R is bounded measurableand for each value of its second argument concave in its first argument Then the function S Linfin rarr Rdefined by S(α) =

983125 1

0h(α(v) v)dK(v) is concave

Proof Fix α1α2 isin Linfin c isin (0 1) and let αc = cα1 + (1minus c)α2 Then

S(αc)minus cS(α1)minus (1minus c)S(α2) =

983133 1

0

983043h(αc(v) v)minus ch(α1(v) v)minus (1minus c)h(α2(v) v)

983044dK(v) ge 0

because concavity of h implies that the integrand is positive for each v and because K isincreasing

Note that for each v u(α(v))f(v)+ κf(v)α(v)2

2is concave in α(v) since its second derivative

is given by f(v)[uprimeprime(α(v)) + κ] which is negative by definition of κ This implies that for eachv each integrand in (A1) is a concave function of α(v) Since Λ is increasing Lemma A3implies that the Lagrangean is concave in α

This implies that the Lagrangean is maximized at α if the Gateaux differential satisfiespartL(ααminus αΛ) le 0 for all α isin A

If Λ(1) = κF (1) (A1) simplifies to

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v)

The Gateaux differential is

partL(ααΛ) =983133 1

0

983059983147uprime(α(v))f(v)minus κF (v) + Λ(v)

983148α(v)

983060dv +

983133 1

0([v minus α(v)]α(v)) d[Λ(v)minus κF (v)]

(A2)

31

=

983133 1

0

983061983133 1

vuprime(α(x))f(x)minus κF (x) + Λ(x)dx

983062dα(v) +

983133 1

0

983061983133 1

v[xminus α(x)]d[Λ(x)minus κF (x)]

983062dα(v)

(A3)

where the second equality obtains using integration by parts

Below we construct increasing and right-continuous Lagrange multipliers that satisfyΛ(1) = κF (1) such that partL(ααminus αΛ) le 0

We begin with the optimality of full delegation

Proof of sufficiency part of Proposition 1 Note that the action function induced by full del-egation is α(v) = v We claim that α maximizes the Lagrangean for the multiplier Λ(v) =

κF (v) minus uprime(v)f(v) for v lt 1 and Λ(1) = κF (1) Note that the multiplier is increasing sinceκF (v) minus uprime(v)f(v) is increasing by assumption and uprime(v) ge 0 The Lagrangean is thereforemaximized at α if partL(ααminusαΛ) le 0 for all α isin A Note that the integrand of the first integralin (A2) is 0 for almost every v by choice of Λ and the second integral is 0 since α(v) = v

We next consider the optimality of no compromise

Proof of sufficiency part of Proposition 2 Note that the action rule induced by no compro-mise satisfies α(v) = 0 for v isin [0 1

2) and α(v) = 1 for v isin [1

2 1] Now suppose for all s isin [0 12)

and t isin (12 1] we have

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus s

and let ψ = inftisin(121](uprime(1) + κ(1minus t))F (t)minusF (12)

tminus12 Define Λ(v) = κF (12)minusψ for v isin [0 1) and

Λ(1) = κF (1)

Let s isin (12 1] Note that integration by parts implies983125 12

svdF (v) = 12F (12)minus sF (s)minus

983125 12

sF (v)dv Since Λ(v) is constant on [0 1) the definition of ψ implies that for any s isin [0 12)

983133 12

s

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 12

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (12)minus F (s)] + 12[Λ(12)minus κF (12)]minus s[Λ(s)minus κF (s)]

=[uprime(0) + κs][F (12)minus F (s)]minus (12minus s)ψ le 0 (A4)

32

Similarly for any t isin (12 1]

983133 t

12

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 t

12

[v minus α(v)]d[Λ(v)minus κF (v)]

=uprime(1)[F (t)minus F (12)] + (tminus 1)[Λ(t)minus κF (t)] + 12[Λ(12)minus κF (12)]

=[uprime(1) + κ(1minus t)][F (t)minus F (12)]minus (tminus 12)ψ ge 0 (A5)

Fix arbitrary α isin A that satisfies α(1) = 1 It follows from (A3) and the definition of α that

partL(ααminus αΛ)

=

983133 1

0

983063983133 1

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 1

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983064d[α(v)minus α(v)]

=

983133 1

0

983077983133 12

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 12

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983078dα(v)

Since α is increasing (A4) and (A5) imply that partL(αα minus αΛ) le 0 Since the optimal actionrule chooses action 1 for type 1 we conclude that α is optimal

Lastly we consider optimality of interval delegation

Proof of sufficiency part of Proposition 3 The induced action function is α(v) = 0 for v lt

clowast2 α(v) = clowast for clowast2 le v le clowast and α(v) = v for v gt clowast

We propose the following multiplier

Λ(v) =

983099983105983105983105983103

983105983105983105983101

κF (clowast2)minus uprime(clowast)F (clowast)minusF (clowast2)clowastminusclowast2 if v lt clowast

κF (v)minus uprime(v)f(v) if clowast le v lt 1

κF (1) if v = 1

Λ is constant on [0 clowast) and it follows from Proposition 3rsquos condition (i) that Λ is increasing on(clowast 1] To see that Λ is increasing at clowast note that condition (ii) holds as an equality for t = clowast

and hence the derivative of the LHS of condition (ii) with respect to t must be negative att = clowast which yields

minusκF (clowast)minus F (clowast2)

clowast2+ uprime(clowast)

f(clowast)clowast2minus (F (clowast)minus F (clowast2))

(clowast2)2le 0

Hence Λ is increasing at clowast It is thus sufficient to show partL(ααminus αΛ) le 0 for all α isin A

33

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

KARLIN S (1968) Total Positivity vol I Stanford California Stanford University Press

KOLOTILIN A (2018) ldquoOptimal Information Disclosure A Linear Programming ApproachrdquoTheoretical Economics 13 607ndash635

KOLOTILIN A AND A ZAPECHELNYUK (2019) ldquoPersuasion meets Delegationrdquo Unpub-lished

KOVAC E AND T MYLOVANOV (2009) ldquoStochastic Mechanisms in Settings Without Mone-tary Transfers The Regular Caserdquo Journal of Economic Theory 144 1373ndash1395

LANDES W M (1971) ldquoAn Economic Analysis of the Courtsrdquo Journal of Law and Economics14 61ndash107

MATTHEWS S A (1989) ldquoVeto Threats Rhetoric in a Bargaining Gamerdquo Quarterly Journal ofEconomics 104 347ndash369

MELUMAD N AND T SHIBANO (1991) ldquoCommunication in Settings with No TransfersrdquoRAND Journal of Economics 22 173ndash198

MILGROM P AND I SEGAL (2002) ldquoEnvelope Theorems for Arbitrary Choice Setsrdquo Econo-metrica 70 583ndash601

MILGROM P AND C SHANNON (1994) ldquoMonotone Comparative Staticsrdquo Econometrica 157ndash180

ORZACH R AND S J SPURR (2008) ldquoLesser-included Offensesrdquo International Review of Lawand Economics 28 239ndash245

PRATT J W (1964) ldquoRisk Aversion in the Small and in the Largerdquo Econometrica 32 122ndash136

ROMER T AND H ROSENTHAL (1978) ldquoPolitical Resource Allocation Controlled Agendasand the Status Quordquo Public Choice 33 27ndash43

mdashmdashmdash (1979) ldquoBureaucrats Versus Voters On the Political Economy of Resource Allocationby Direct Democracyrdquo Quarterly Journal of Economics 93 563ndash587

ROYDEN H AND P FITZPATRICK (2010) Real Analysis Prentice-Hall Inc 4th ed

VOLDEN C (2002) ldquoA Formal Model of the Politics of Delegation in a Separation of PowersSystemrdquo American Journal of Political Science 46 111ndash133

ZAPECHELNYUK A (2019) ldquoOptimal Quality Certificationrdquo American Economic Review In-sights 2 161ndash176

28

Appendices

A Proofs of Propositions 1 2 and 3

In Appendix A we assume the support of the type distribution F is [0 1] This is withoutloss (even among stochastic mechanisms) because it is always optimal for Proposer to chooseaction 1 for types above 1 and given the outside option to choose action 0 for types below 0

A1 Sufficient Conditions

For convenience we recall Proposerrsquos problem (P)

maxmisinS

983133 1

0

Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046minus

983133 v

0

Em(x)[a]dx = 0 forallv isin [0 1] (IC-env)

We also recall the relaxed problem (R)

maxαisinA

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx ge 0 forallv isin [0 1]

We show first that any IC solution to the relaxed problem also solves problem (P)

Lemma A1 Suppose αlowast isin A solves problem (R) and is incentive compatible Then αlowast also solves(P)

Proof To obtain a contradiction let m isin S be feasible for (P) and suppose it achieves a strictlyhigher objective value in (P) than αlowast Define α isin A by setting α(v) = Em(v)[a] for each vThen Jensenrsquos inequality implies Em(v) [av minus a22] ge vα(v)minusα(v)22 whereas

983125 v

0Em(x)[a]dx =

983125 v

0α(x)ds Hence feasibility of m in (P) implies feasibility of α in (R) Moreover

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v)

ge983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(x)[a]dx

983062dF (v)

=

983133 1

0

Em(v)[u(a)]dF (v)

29

gt

983133 1

0

u(αlowast(v))dF (v)

=

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(x)dx

983064983062dF (v)

where the first inequality holds because the first line is a concave functional (by the defini-tion of κ equiv infaisin[01) minusuprimeprime(a)) the first equality holds because m is feasible in (P) the secondinequality holds because of our assumption that m achieves a strictly higher value than αlowastand the final equality holds because αlowast being IC implies it is feasible in (P) Therefore αlowast isnot optimal in (R) a contradiction

To show that a given delegation set solves the relaxed problem we construct Lagrangemultipliers and show that the induced action rule maximizes the following Lagrangean Givenα isin A and an increasing and right-continuous function Λ(v) let the Lagrangean be given by

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus κf(v)

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983062dΛ(v)

=

983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v) +

983133 1

0

α(v)dv[κF (1)minus Λ(1)] (A1)

where the second equality follows from integration by parts28

Lemma A2 Let αlowast be induced by a delegation set29 Suppose there is an increasing and right-continuous function Λ such that L(αlowastΛ) ge L(αΛ) for all α isin A Then α solves problem (R)

Proof Under the conditions stated in the lemma Theorem 1 in Amador and Bagwell (2013)implies that αlowast solves the relaxed problem To see this let their f be the negative of theobjective function in (R) X and Z be the vector space of bounded measurable functions andΩ = A let P = z isin Z|forallv isin [0 1] z(v) ge 0 and G be the negative of the left-hand side of theconstraint in (R) Define the linear function T Z rarr R by T (z) =

983125 1

0z(v)dΛ(v) (which is well-

defined since Λ corresponds to a finite measure (Royden and Fitzpatrick 2010 p 437) and

28983125 v

0α(s)ds is continuous and κF (v) minus Λ(v) has bounded variation as the difference of two increasing func-

tions Hence the Riemann-Stieltjes integral983125 1

0

983125 v

0α(s)dsd[κF (v)minus Λ(v)] exists and integration by parts is valid

29 That is there is some delegation set A such that αlowast(v) is an action in A cup 0 that type v prefers the most

30

the integral therefore exists because z is bounded and measurable (Royden and Fitzpatrick2010 p 375)) and observe that for all z isin P T (z) ge 0 and hence condition (i) holds Sinceαlowast is incentive compatible minusG(αlowast) isin P (condition (iii) holds) and T (G(αlowast)) = 0 (condition(iv) holds) Since αlowast maximizes the Lagrangean by assumption condition (ii) holds and weconclude that αlowast solves problem (R)

The follow observation will allow us to establish that the Lagrangean is a concave func-tional of α if Λ is increasing

Lemma A3 Suppose K is right-continuous and increasing and h R2 rarr R is bounded measurableand for each value of its second argument concave in its first argument Then the function S Linfin rarr Rdefined by S(α) =

983125 1

0h(α(v) v)dK(v) is concave

Proof Fix α1α2 isin Linfin c isin (0 1) and let αc = cα1 + (1minus c)α2 Then

S(αc)minus cS(α1)minus (1minus c)S(α2) =

983133 1

0

983043h(αc(v) v)minus ch(α1(v) v)minus (1minus c)h(α2(v) v)

983044dK(v) ge 0

because concavity of h implies that the integrand is positive for each v and because K isincreasing

Note that for each v u(α(v))f(v)+ κf(v)α(v)2

2is concave in α(v) since its second derivative

is given by f(v)[uprimeprime(α(v)) + κ] which is negative by definition of κ This implies that for eachv each integrand in (A1) is a concave function of α(v) Since Λ is increasing Lemma A3implies that the Lagrangean is concave in α

This implies that the Lagrangean is maximized at α if the Gateaux differential satisfiespartL(ααminus αΛ) le 0 for all α isin A

If Λ(1) = κF (1) (A1) simplifies to

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v)

The Gateaux differential is

partL(ααΛ) =983133 1

0

983059983147uprime(α(v))f(v)minus κF (v) + Λ(v)

983148α(v)

983060dv +

983133 1

0([v minus α(v)]α(v)) d[Λ(v)minus κF (v)]

(A2)

31

=

983133 1

0

983061983133 1

vuprime(α(x))f(x)minus κF (x) + Λ(x)dx

983062dα(v) +

983133 1

0

983061983133 1

v[xminus α(x)]d[Λ(x)minus κF (x)]

983062dα(v)

(A3)

where the second equality obtains using integration by parts

Below we construct increasing and right-continuous Lagrange multipliers that satisfyΛ(1) = κF (1) such that partL(ααminus αΛ) le 0

We begin with the optimality of full delegation

Proof of sufficiency part of Proposition 1 Note that the action function induced by full del-egation is α(v) = v We claim that α maximizes the Lagrangean for the multiplier Λ(v) =

κF (v) minus uprime(v)f(v) for v lt 1 and Λ(1) = κF (1) Note that the multiplier is increasing sinceκF (v) minus uprime(v)f(v) is increasing by assumption and uprime(v) ge 0 The Lagrangean is thereforemaximized at α if partL(ααminusαΛ) le 0 for all α isin A Note that the integrand of the first integralin (A2) is 0 for almost every v by choice of Λ and the second integral is 0 since α(v) = v

We next consider the optimality of no compromise

Proof of sufficiency part of Proposition 2 Note that the action rule induced by no compro-mise satisfies α(v) = 0 for v isin [0 1

2) and α(v) = 1 for v isin [1

2 1] Now suppose for all s isin [0 12)

and t isin (12 1] we have

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus s

and let ψ = inftisin(121](uprime(1) + κ(1minus t))F (t)minusF (12)

tminus12 Define Λ(v) = κF (12)minusψ for v isin [0 1) and

Λ(1) = κF (1)

Let s isin (12 1] Note that integration by parts implies983125 12

svdF (v) = 12F (12)minus sF (s)minus

983125 12

sF (v)dv Since Λ(v) is constant on [0 1) the definition of ψ implies that for any s isin [0 12)

983133 12

s

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 12

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (12)minus F (s)] + 12[Λ(12)minus κF (12)]minus s[Λ(s)minus κF (s)]

=[uprime(0) + κs][F (12)minus F (s)]minus (12minus s)ψ le 0 (A4)

32

Similarly for any t isin (12 1]

983133 t

12

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 t

12

[v minus α(v)]d[Λ(v)minus κF (v)]

=uprime(1)[F (t)minus F (12)] + (tminus 1)[Λ(t)minus κF (t)] + 12[Λ(12)minus κF (12)]

=[uprime(1) + κ(1minus t)][F (t)minus F (12)]minus (tminus 12)ψ ge 0 (A5)

Fix arbitrary α isin A that satisfies α(1) = 1 It follows from (A3) and the definition of α that

partL(ααminus αΛ)

=

983133 1

0

983063983133 1

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 1

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983064d[α(v)minus α(v)]

=

983133 1

0

983077983133 12

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 12

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983078dα(v)

Since α is increasing (A4) and (A5) imply that partL(αα minus αΛ) le 0 Since the optimal actionrule chooses action 1 for type 1 we conclude that α is optimal

Lastly we consider optimality of interval delegation

Proof of sufficiency part of Proposition 3 The induced action function is α(v) = 0 for v lt

clowast2 α(v) = clowast for clowast2 le v le clowast and α(v) = v for v gt clowast

We propose the following multiplier

Λ(v) =

983099983105983105983105983103

983105983105983105983101

κF (clowast2)minus uprime(clowast)F (clowast)minusF (clowast2)clowastminusclowast2 if v lt clowast

κF (v)minus uprime(v)f(v) if clowast le v lt 1

κF (1) if v = 1

Λ is constant on [0 clowast) and it follows from Proposition 3rsquos condition (i) that Λ is increasing on(clowast 1] To see that Λ is increasing at clowast note that condition (ii) holds as an equality for t = clowast

and hence the derivative of the LHS of condition (ii) with respect to t must be negative att = clowast which yields

minusκF (clowast)minus F (clowast2)

clowast2+ uprime(clowast)

f(clowast)clowast2minus (F (clowast)minus F (clowast2))

(clowast2)2le 0

Hence Λ is increasing at clowast It is thus sufficient to show partL(ααminus αΛ) le 0 for all α isin A

33

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

Appendices

A Proofs of Propositions 1 2 and 3

In Appendix A we assume the support of the type distribution F is [0 1] This is withoutloss (even among stochastic mechanisms) because it is always optimal for Proposer to chooseaction 1 for types above 1 and given the outside option to choose action 0 for types below 0

A1 Sufficient Conditions

For convenience we recall Proposerrsquos problem (P)

maxmisinS

983133 1

0

Em(v)[u(a)]dF (v) (P)

st Em(v)

983045av minus a22

983046minus

983133 v

0

Em(x)[a]dx = 0 forallv isin [0 1] (IC-env)

We also recall the relaxed problem (R)

maxαisinA

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v) (R)

st vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx ge 0 forallv isin [0 1]

We show first that any IC solution to the relaxed problem also solves problem (P)

Lemma A1 Suppose αlowast isin A solves problem (R) and is incentive compatible Then αlowast also solves(P)

Proof To obtain a contradiction let m isin S be feasible for (P) and suppose it achieves a strictlyhigher objective value in (P) than αlowast Define α isin A by setting α(v) = Em(v)[a] for each vThen Jensenrsquos inequality implies Em(v) [av minus a22] ge vα(v)minusα(v)22 whereas

983125 v

0Em(x)[a]dx =

983125 v

0α(x)ds Hence feasibility of m in (P) implies feasibility of α in (R) Moreover

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dF (v)

ge983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(x)[a]dx

983062dF (v)

=

983133 1

0

Em(v)[u(a)]dF (v)

29

gt

983133 1

0

u(αlowast(v))dF (v)

=

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(x)dx

983064983062dF (v)

where the first inequality holds because the first line is a concave functional (by the defini-tion of κ equiv infaisin[01) minusuprimeprime(a)) the first equality holds because m is feasible in (P) the secondinequality holds because of our assumption that m achieves a strictly higher value than αlowastand the final equality holds because αlowast being IC implies it is feasible in (P) Therefore αlowast isnot optimal in (R) a contradiction

To show that a given delegation set solves the relaxed problem we construct Lagrangemultipliers and show that the induced action rule maximizes the following Lagrangean Givenα isin A and an increasing and right-continuous function Λ(v) let the Lagrangean be given by

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus κf(v)

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983062dΛ(v)

=

983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v) +

983133 1

0

α(v)dv[κF (1)minus Λ(1)] (A1)

where the second equality follows from integration by parts28

Lemma A2 Let αlowast be induced by a delegation set29 Suppose there is an increasing and right-continuous function Λ such that L(αlowastΛ) ge L(αΛ) for all α isin A Then α solves problem (R)

Proof Under the conditions stated in the lemma Theorem 1 in Amador and Bagwell (2013)implies that αlowast solves the relaxed problem To see this let their f be the negative of theobjective function in (R) X and Z be the vector space of bounded measurable functions andΩ = A let P = z isin Z|forallv isin [0 1] z(v) ge 0 and G be the negative of the left-hand side of theconstraint in (R) Define the linear function T Z rarr R by T (z) =

983125 1

0z(v)dΛ(v) (which is well-

defined since Λ corresponds to a finite measure (Royden and Fitzpatrick 2010 p 437) and

28983125 v

0α(s)ds is continuous and κF (v) minus Λ(v) has bounded variation as the difference of two increasing func-

tions Hence the Riemann-Stieltjes integral983125 1

0

983125 v

0α(s)dsd[κF (v)minus Λ(v)] exists and integration by parts is valid

29 That is there is some delegation set A such that αlowast(v) is an action in A cup 0 that type v prefers the most

30

the integral therefore exists because z is bounded and measurable (Royden and Fitzpatrick2010 p 375)) and observe that for all z isin P T (z) ge 0 and hence condition (i) holds Sinceαlowast is incentive compatible minusG(αlowast) isin P (condition (iii) holds) and T (G(αlowast)) = 0 (condition(iv) holds) Since αlowast maximizes the Lagrangean by assumption condition (ii) holds and weconclude that αlowast solves problem (R)

The follow observation will allow us to establish that the Lagrangean is a concave func-tional of α if Λ is increasing

Lemma A3 Suppose K is right-continuous and increasing and h R2 rarr R is bounded measurableand for each value of its second argument concave in its first argument Then the function S Linfin rarr Rdefined by S(α) =

983125 1

0h(α(v) v)dK(v) is concave

Proof Fix α1α2 isin Linfin c isin (0 1) and let αc = cα1 + (1minus c)α2 Then

S(αc)minus cS(α1)minus (1minus c)S(α2) =

983133 1

0

983043h(αc(v) v)minus ch(α1(v) v)minus (1minus c)h(α2(v) v)

983044dK(v) ge 0

because concavity of h implies that the integrand is positive for each v and because K isincreasing

Note that for each v u(α(v))f(v)+ κf(v)α(v)2

2is concave in α(v) since its second derivative

is given by f(v)[uprimeprime(α(v)) + κ] which is negative by definition of κ This implies that for eachv each integrand in (A1) is a concave function of α(v) Since Λ is increasing Lemma A3implies that the Lagrangean is concave in α

This implies that the Lagrangean is maximized at α if the Gateaux differential satisfiespartL(ααminus αΛ) le 0 for all α isin A

If Λ(1) = κF (1) (A1) simplifies to

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v)

The Gateaux differential is

partL(ααΛ) =983133 1

0

983059983147uprime(α(v))f(v)minus κF (v) + Λ(v)

983148α(v)

983060dv +

983133 1

0([v minus α(v)]α(v)) d[Λ(v)minus κF (v)]

(A2)

31

=

983133 1

0

983061983133 1

vuprime(α(x))f(x)minus κF (x) + Λ(x)dx

983062dα(v) +

983133 1

0

983061983133 1

v[xminus α(x)]d[Λ(x)minus κF (x)]

983062dα(v)

(A3)

where the second equality obtains using integration by parts

Below we construct increasing and right-continuous Lagrange multipliers that satisfyΛ(1) = κF (1) such that partL(ααminus αΛ) le 0

We begin with the optimality of full delegation

Proof of sufficiency part of Proposition 1 Note that the action function induced by full del-egation is α(v) = v We claim that α maximizes the Lagrangean for the multiplier Λ(v) =

κF (v) minus uprime(v)f(v) for v lt 1 and Λ(1) = κF (1) Note that the multiplier is increasing sinceκF (v) minus uprime(v)f(v) is increasing by assumption and uprime(v) ge 0 The Lagrangean is thereforemaximized at α if partL(ααminusαΛ) le 0 for all α isin A Note that the integrand of the first integralin (A2) is 0 for almost every v by choice of Λ and the second integral is 0 since α(v) = v

We next consider the optimality of no compromise

Proof of sufficiency part of Proposition 2 Note that the action rule induced by no compro-mise satisfies α(v) = 0 for v isin [0 1

2) and α(v) = 1 for v isin [1

2 1] Now suppose for all s isin [0 12)

and t isin (12 1] we have

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus s

and let ψ = inftisin(121](uprime(1) + κ(1minus t))F (t)minusF (12)

tminus12 Define Λ(v) = κF (12)minusψ for v isin [0 1) and

Λ(1) = κF (1)

Let s isin (12 1] Note that integration by parts implies983125 12

svdF (v) = 12F (12)minus sF (s)minus

983125 12

sF (v)dv Since Λ(v) is constant on [0 1) the definition of ψ implies that for any s isin [0 12)

983133 12

s

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 12

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (12)minus F (s)] + 12[Λ(12)minus κF (12)]minus s[Λ(s)minus κF (s)]

=[uprime(0) + κs][F (12)minus F (s)]minus (12minus s)ψ le 0 (A4)

32

Similarly for any t isin (12 1]

983133 t

12

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 t

12

[v minus α(v)]d[Λ(v)minus κF (v)]

=uprime(1)[F (t)minus F (12)] + (tminus 1)[Λ(t)minus κF (t)] + 12[Λ(12)minus κF (12)]

=[uprime(1) + κ(1minus t)][F (t)minus F (12)]minus (tminus 12)ψ ge 0 (A5)

Fix arbitrary α isin A that satisfies α(1) = 1 It follows from (A3) and the definition of α that

partL(ααminus αΛ)

=

983133 1

0

983063983133 1

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 1

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983064d[α(v)minus α(v)]

=

983133 1

0

983077983133 12

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 12

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983078dα(v)

Since α is increasing (A4) and (A5) imply that partL(αα minus αΛ) le 0 Since the optimal actionrule chooses action 1 for type 1 we conclude that α is optimal

Lastly we consider optimality of interval delegation

Proof of sufficiency part of Proposition 3 The induced action function is α(v) = 0 for v lt

clowast2 α(v) = clowast for clowast2 le v le clowast and α(v) = v for v gt clowast

We propose the following multiplier

Λ(v) =

983099983105983105983105983103

983105983105983105983101

κF (clowast2)minus uprime(clowast)F (clowast)minusF (clowast2)clowastminusclowast2 if v lt clowast

κF (v)minus uprime(v)f(v) if clowast le v lt 1

κF (1) if v = 1

Λ is constant on [0 clowast) and it follows from Proposition 3rsquos condition (i) that Λ is increasing on(clowast 1] To see that Λ is increasing at clowast note that condition (ii) holds as an equality for t = clowast

and hence the derivative of the LHS of condition (ii) with respect to t must be negative att = clowast which yields

minusκF (clowast)minus F (clowast2)

clowast2+ uprime(clowast)

f(clowast)clowast2minus (F (clowast)minus F (clowast2))

(clowast2)2le 0

Hence Λ is increasing at clowast It is thus sufficient to show partL(ααminus αΛ) le 0 for all α isin A

33

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

gt

983133 1

0

u(αlowast(v))dF (v)

=

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(x)dx

983064983062dF (v)

where the first inequality holds because the first line is a concave functional (by the defini-tion of κ equiv infaisin[01) minusuprimeprime(a)) the first equality holds because m is feasible in (P) the secondinequality holds because of our assumption that m achieves a strictly higher value than αlowastand the final equality holds because αlowast being IC implies it is feasible in (P) Therefore αlowast isnot optimal in (R) a contradiction

To show that a given delegation set solves the relaxed problem we construct Lagrangemultipliers and show that the induced action rule maximizes the following Lagrangean Givenα isin A and an increasing and right-continuous function Λ(v) let the Lagrangean be given by

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus κf(v)

983063vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2minus

983133 v

0

α(x)dx

983062dΛ(v)

=

983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v) +

983133 1

0

α(v)dv[κF (1)minus Λ(1)] (A1)

where the second equality follows from integration by parts28

Lemma A2 Let αlowast be induced by a delegation set29 Suppose there is an increasing and right-continuous function Λ such that L(αlowastΛ) ge L(αΛ) for all α isin A Then α solves problem (R)

Proof Under the conditions stated in the lemma Theorem 1 in Amador and Bagwell (2013)implies that αlowast solves the relaxed problem To see this let their f be the negative of theobjective function in (R) X and Z be the vector space of bounded measurable functions andΩ = A let P = z isin Z|forallv isin [0 1] z(v) ge 0 and G be the negative of the left-hand side of theconstraint in (R) Define the linear function T Z rarr R by T (z) =

983125 1

0z(v)dΛ(v) (which is well-

defined since Λ corresponds to a finite measure (Royden and Fitzpatrick 2010 p 437) and

28983125 v

0α(s)ds is continuous and κF (v) minus Λ(v) has bounded variation as the difference of two increasing func-

tions Hence the Riemann-Stieltjes integral983125 1

0

983125 v

0α(s)dsd[κF (v)minus Λ(v)] exists and integration by parts is valid

29 That is there is some delegation set A such that αlowast(v) is an action in A cup 0 that type v prefers the most

30

the integral therefore exists because z is bounded and measurable (Royden and Fitzpatrick2010 p 375)) and observe that for all z isin P T (z) ge 0 and hence condition (i) holds Sinceαlowast is incentive compatible minusG(αlowast) isin P (condition (iii) holds) and T (G(αlowast)) = 0 (condition(iv) holds) Since αlowast maximizes the Lagrangean by assumption condition (ii) holds and weconclude that αlowast solves problem (R)

The follow observation will allow us to establish that the Lagrangean is a concave func-tional of α if Λ is increasing

Lemma A3 Suppose K is right-continuous and increasing and h R2 rarr R is bounded measurableand for each value of its second argument concave in its first argument Then the function S Linfin rarr Rdefined by S(α) =

983125 1

0h(α(v) v)dK(v) is concave

Proof Fix α1α2 isin Linfin c isin (0 1) and let αc = cα1 + (1minus c)α2 Then

S(αc)minus cS(α1)minus (1minus c)S(α2) =

983133 1

0

983043h(αc(v) v)minus ch(α1(v) v)minus (1minus c)h(α2(v) v)

983044dK(v) ge 0

because concavity of h implies that the integrand is positive for each v and because K isincreasing

Note that for each v u(α(v))f(v)+ κf(v)α(v)2

2is concave in α(v) since its second derivative

is given by f(v)[uprimeprime(α(v)) + κ] which is negative by definition of κ This implies that for eachv each integrand in (A1) is a concave function of α(v) Since Λ is increasing Lemma A3implies that the Lagrangean is concave in α

This implies that the Lagrangean is maximized at α if the Gateaux differential satisfiespartL(ααminus αΛ) le 0 for all α isin A

If Λ(1) = κF (1) (A1) simplifies to

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v)

The Gateaux differential is

partL(ααΛ) =983133 1

0

983059983147uprime(α(v))f(v)minus κF (v) + Λ(v)

983148α(v)

983060dv +

983133 1

0([v minus α(v)]α(v)) d[Λ(v)minus κF (v)]

(A2)

31

=

983133 1

0

983061983133 1

vuprime(α(x))f(x)minus κF (x) + Λ(x)dx

983062dα(v) +

983133 1

0

983061983133 1

v[xminus α(x)]d[Λ(x)minus κF (x)]

983062dα(v)

(A3)

where the second equality obtains using integration by parts

Below we construct increasing and right-continuous Lagrange multipliers that satisfyΛ(1) = κF (1) such that partL(ααminus αΛ) le 0

We begin with the optimality of full delegation

Proof of sufficiency part of Proposition 1 Note that the action function induced by full del-egation is α(v) = v We claim that α maximizes the Lagrangean for the multiplier Λ(v) =

κF (v) minus uprime(v)f(v) for v lt 1 and Λ(1) = κF (1) Note that the multiplier is increasing sinceκF (v) minus uprime(v)f(v) is increasing by assumption and uprime(v) ge 0 The Lagrangean is thereforemaximized at α if partL(ααminusαΛ) le 0 for all α isin A Note that the integrand of the first integralin (A2) is 0 for almost every v by choice of Λ and the second integral is 0 since α(v) = v

We next consider the optimality of no compromise

Proof of sufficiency part of Proposition 2 Note that the action rule induced by no compro-mise satisfies α(v) = 0 for v isin [0 1

2) and α(v) = 1 for v isin [1

2 1] Now suppose for all s isin [0 12)

and t isin (12 1] we have

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus s

and let ψ = inftisin(121](uprime(1) + κ(1minus t))F (t)minusF (12)

tminus12 Define Λ(v) = κF (12)minusψ for v isin [0 1) and

Λ(1) = κF (1)

Let s isin (12 1] Note that integration by parts implies983125 12

svdF (v) = 12F (12)minus sF (s)minus

983125 12

sF (v)dv Since Λ(v) is constant on [0 1) the definition of ψ implies that for any s isin [0 12)

983133 12

s

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 12

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (12)minus F (s)] + 12[Λ(12)minus κF (12)]minus s[Λ(s)minus κF (s)]

=[uprime(0) + κs][F (12)minus F (s)]minus (12minus s)ψ le 0 (A4)

32

Similarly for any t isin (12 1]

983133 t

12

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 t

12

[v minus α(v)]d[Λ(v)minus κF (v)]

=uprime(1)[F (t)minus F (12)] + (tminus 1)[Λ(t)minus κF (t)] + 12[Λ(12)minus κF (12)]

=[uprime(1) + κ(1minus t)][F (t)minus F (12)]minus (tminus 12)ψ ge 0 (A5)

Fix arbitrary α isin A that satisfies α(1) = 1 It follows from (A3) and the definition of α that

partL(ααminus αΛ)

=

983133 1

0

983063983133 1

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 1

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983064d[α(v)minus α(v)]

=

983133 1

0

983077983133 12

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 12

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983078dα(v)

Since α is increasing (A4) and (A5) imply that partL(αα minus αΛ) le 0 Since the optimal actionrule chooses action 1 for type 1 we conclude that α is optimal

Lastly we consider optimality of interval delegation

Proof of sufficiency part of Proposition 3 The induced action function is α(v) = 0 for v lt

clowast2 α(v) = clowast for clowast2 le v le clowast and α(v) = v for v gt clowast

We propose the following multiplier

Λ(v) =

983099983105983105983105983103

983105983105983105983101

κF (clowast2)minus uprime(clowast)F (clowast)minusF (clowast2)clowastminusclowast2 if v lt clowast

κF (v)minus uprime(v)f(v) if clowast le v lt 1

κF (1) if v = 1

Λ is constant on [0 clowast) and it follows from Proposition 3rsquos condition (i) that Λ is increasing on(clowast 1] To see that Λ is increasing at clowast note that condition (ii) holds as an equality for t = clowast

and hence the derivative of the LHS of condition (ii) with respect to t must be negative att = clowast which yields

minusκF (clowast)minus F (clowast2)

clowast2+ uprime(clowast)

f(clowast)clowast2minus (F (clowast)minus F (clowast2))

(clowast2)2le 0

Hence Λ is increasing at clowast It is thus sufficient to show partL(ααminus αΛ) le 0 for all α isin A

33

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

the integral therefore exists because z is bounded and measurable (Royden and Fitzpatrick2010 p 375)) and observe that for all z isin P T (z) ge 0 and hence condition (i) holds Sinceαlowast is incentive compatible minusG(αlowast) isin P (condition (iii) holds) and T (G(αlowast)) = 0 (condition(iv) holds) Since αlowast maximizes the Lagrangean by assumption condition (ii) holds and weconclude that αlowast solves problem (R)

The follow observation will allow us to establish that the Lagrangean is a concave func-tional of α if Λ is increasing

Lemma A3 Suppose K is right-continuous and increasing and h R2 rarr R is bounded measurableand for each value of its second argument concave in its first argument Then the function S Linfin rarr Rdefined by S(α) =

983125 1

0h(α(v) v)dK(v) is concave

Proof Fix α1α2 isin Linfin c isin (0 1) and let αc = cα1 + (1minus c)α2 Then

S(αc)minus cS(α1)minus (1minus c)S(α2) =

983133 1

0

983043h(αc(v) v)minus ch(α1(v) v)minus (1minus c)h(α2(v) v)

983044dK(v) ge 0

because concavity of h implies that the integrand is positive for each v and because K isincreasing

Note that for each v u(α(v))f(v)+ κf(v)α(v)2

2is concave in α(v) since its second derivative

is given by f(v)[uprimeprime(α(v)) + κ] which is negative by definition of κ This implies that for eachv each integrand in (A1) is a concave function of α(v) Since Λ is increasing Lemma A3implies that the Lagrangean is concave in α

This implies that the Lagrangean is maximized at α if the Gateaux differential satisfiespartL(ααminus αΛ) le 0 for all α isin A

If Λ(1) = κF (1) (A1) simplifies to

L(αΛ) =983133 1

0

983061u(α(v))f(v)minus α(v)[κF (v)minus Λ(v)]minus κf(v)

983063vα(v)minus α(v)2

2

983064983062dv

+

983133 1

0

983061vα(v)minus α(v)2

2

983062dΛ(v)

The Gateaux differential is

partL(ααΛ) =983133 1

0

983059983147uprime(α(v))f(v)minus κF (v) + Λ(v)

983148α(v)

983060dv +

983133 1

0([v minus α(v)]α(v)) d[Λ(v)minus κF (v)]

(A2)

31

=

983133 1

0

983061983133 1

vuprime(α(x))f(x)minus κF (x) + Λ(x)dx

983062dα(v) +

983133 1

0

983061983133 1

v[xminus α(x)]d[Λ(x)minus κF (x)]

983062dα(v)

(A3)

where the second equality obtains using integration by parts

Below we construct increasing and right-continuous Lagrange multipliers that satisfyΛ(1) = κF (1) such that partL(ααminus αΛ) le 0

We begin with the optimality of full delegation

Proof of sufficiency part of Proposition 1 Note that the action function induced by full del-egation is α(v) = v We claim that α maximizes the Lagrangean for the multiplier Λ(v) =

κF (v) minus uprime(v)f(v) for v lt 1 and Λ(1) = κF (1) Note that the multiplier is increasing sinceκF (v) minus uprime(v)f(v) is increasing by assumption and uprime(v) ge 0 The Lagrangean is thereforemaximized at α if partL(ααminusαΛ) le 0 for all α isin A Note that the integrand of the first integralin (A2) is 0 for almost every v by choice of Λ and the second integral is 0 since α(v) = v

We next consider the optimality of no compromise

Proof of sufficiency part of Proposition 2 Note that the action rule induced by no compro-mise satisfies α(v) = 0 for v isin [0 1

2) and α(v) = 1 for v isin [1

2 1] Now suppose for all s isin [0 12)

and t isin (12 1] we have

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus s

and let ψ = inftisin(121](uprime(1) + κ(1minus t))F (t)minusF (12)

tminus12 Define Λ(v) = κF (12)minusψ for v isin [0 1) and

Λ(1) = κF (1)

Let s isin (12 1] Note that integration by parts implies983125 12

svdF (v) = 12F (12)minus sF (s)minus

983125 12

sF (v)dv Since Λ(v) is constant on [0 1) the definition of ψ implies that for any s isin [0 12)

983133 12

s

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 12

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (12)minus F (s)] + 12[Λ(12)minus κF (12)]minus s[Λ(s)minus κF (s)]

=[uprime(0) + κs][F (12)minus F (s)]minus (12minus s)ψ le 0 (A4)

32

Similarly for any t isin (12 1]

983133 t

12

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 t

12

[v minus α(v)]d[Λ(v)minus κF (v)]

=uprime(1)[F (t)minus F (12)] + (tminus 1)[Λ(t)minus κF (t)] + 12[Λ(12)minus κF (12)]

=[uprime(1) + κ(1minus t)][F (t)minus F (12)]minus (tminus 12)ψ ge 0 (A5)

Fix arbitrary α isin A that satisfies α(1) = 1 It follows from (A3) and the definition of α that

partL(ααminus αΛ)

=

983133 1

0

983063983133 1

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 1

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983064d[α(v)minus α(v)]

=

983133 1

0

983077983133 12

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 12

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983078dα(v)

Since α is increasing (A4) and (A5) imply that partL(αα minus αΛ) le 0 Since the optimal actionrule chooses action 1 for type 1 we conclude that α is optimal

Lastly we consider optimality of interval delegation

Proof of sufficiency part of Proposition 3 The induced action function is α(v) = 0 for v lt

clowast2 α(v) = clowast for clowast2 le v le clowast and α(v) = v for v gt clowast

We propose the following multiplier

Λ(v) =

983099983105983105983105983103

983105983105983105983101

κF (clowast2)minus uprime(clowast)F (clowast)minusF (clowast2)clowastminusclowast2 if v lt clowast

κF (v)minus uprime(v)f(v) if clowast le v lt 1

κF (1) if v = 1

Λ is constant on [0 clowast) and it follows from Proposition 3rsquos condition (i) that Λ is increasing on(clowast 1] To see that Λ is increasing at clowast note that condition (ii) holds as an equality for t = clowast

and hence the derivative of the LHS of condition (ii) with respect to t must be negative att = clowast which yields

minusκF (clowast)minus F (clowast2)

clowast2+ uprime(clowast)

f(clowast)clowast2minus (F (clowast)minus F (clowast2))

(clowast2)2le 0

Hence Λ is increasing at clowast It is thus sufficient to show partL(ααminus αΛ) le 0 for all α isin A

33

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

=

983133 1

0

983061983133 1

vuprime(α(x))f(x)minus κF (x) + Λ(x)dx

983062dα(v) +

983133 1

0

983061983133 1

v[xminus α(x)]d[Λ(x)minus κF (x)]

983062dα(v)

(A3)

where the second equality obtains using integration by parts

Below we construct increasing and right-continuous Lagrange multipliers that satisfyΛ(1) = κF (1) such that partL(ααminus αΛ) le 0

We begin with the optimality of full delegation

Proof of sufficiency part of Proposition 1 Note that the action function induced by full del-egation is α(v) = v We claim that α maximizes the Lagrangean for the multiplier Λ(v) =

κF (v) minus uprime(v)f(v) for v lt 1 and Λ(1) = κF (1) Note that the multiplier is increasing sinceκF (v) minus uprime(v)f(v) is increasing by assumption and uprime(v) ge 0 The Lagrangean is thereforemaximized at α if partL(ααminusαΛ) le 0 for all α isin A Note that the integrand of the first integralin (A2) is 0 for almost every v by choice of Λ and the second integral is 0 since α(v) = v

We next consider the optimality of no compromise

Proof of sufficiency part of Proposition 2 Note that the action rule induced by no compro-mise satisfies α(v) = 0 for v isin [0 1

2) and α(v) = 1 for v isin [1

2 1] Now suppose for all s isin [0 12)

and t isin (12 1] we have

(uprime(1) + κ(1minus t))F (t)minus F (12)

tminus 12ge (uprime(0)minus κs)

F (12)minus F (s)

12minus s

and let ψ = inftisin(121](uprime(1) + κ(1minus t))F (t)minusF (12)

tminus12 Define Λ(v) = κF (12)minusψ for v isin [0 1) and

Λ(1) = κF (1)

Let s isin (12 1] Note that integration by parts implies983125 12

svdF (v) = 12F (12)minus sF (s)minus

983125 12

sF (v)dv Since Λ(v) is constant on [0 1) the definition of ψ implies that for any s isin [0 12)

983133 12

s

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 12

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (12)minus F (s)] + 12[Λ(12)minus κF (12)]minus s[Λ(s)minus κF (s)]

=[uprime(0) + κs][F (12)minus F (s)]minus (12minus s)ψ le 0 (A4)

32

Similarly for any t isin (12 1]

983133 t

12

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 t

12

[v minus α(v)]d[Λ(v)minus κF (v)]

=uprime(1)[F (t)minus F (12)] + (tminus 1)[Λ(t)minus κF (t)] + 12[Λ(12)minus κF (12)]

=[uprime(1) + κ(1minus t)][F (t)minus F (12)]minus (tminus 12)ψ ge 0 (A5)

Fix arbitrary α isin A that satisfies α(1) = 1 It follows from (A3) and the definition of α that

partL(ααminus αΛ)

=

983133 1

0

983063983133 1

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 1

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983064d[α(v)minus α(v)]

=

983133 1

0

983077983133 12

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 12

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983078dα(v)

Since α is increasing (A4) and (A5) imply that partL(αα minus αΛ) le 0 Since the optimal actionrule chooses action 1 for type 1 we conclude that α is optimal

Lastly we consider optimality of interval delegation

Proof of sufficiency part of Proposition 3 The induced action function is α(v) = 0 for v lt

clowast2 α(v) = clowast for clowast2 le v le clowast and α(v) = v for v gt clowast

We propose the following multiplier

Λ(v) =

983099983105983105983105983103

983105983105983105983101

κF (clowast2)minus uprime(clowast)F (clowast)minusF (clowast2)clowastminusclowast2 if v lt clowast

κF (v)minus uprime(v)f(v) if clowast le v lt 1

κF (1) if v = 1

Λ is constant on [0 clowast) and it follows from Proposition 3rsquos condition (i) that Λ is increasing on(clowast 1] To see that Λ is increasing at clowast note that condition (ii) holds as an equality for t = clowast

and hence the derivative of the LHS of condition (ii) with respect to t must be negative att = clowast which yields

minusκF (clowast)minus F (clowast2)

clowast2+ uprime(clowast)

f(clowast)clowast2minus (F (clowast)minus F (clowast2))

(clowast2)2le 0

Hence Λ is increasing at clowast It is thus sufficient to show partL(ααminus αΛ) le 0 for all α isin A

33

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

Similarly for any t isin (12 1]

983133 t

12

uprime(α(v))f(v)minus κF (v) + Λ(v)dv +

983133 t

12

[v minus α(v)]d[Λ(v)minus κF (v)]

=uprime(1)[F (t)minus F (12)] + (tminus 1)[Λ(t)minus κF (t)] + 12[Λ(12)minus κF (12)]

=[uprime(1) + κ(1minus t)][F (t)minus F (12)]minus (tminus 12)ψ ge 0 (A5)

Fix arbitrary α isin A that satisfies α(1) = 1 It follows from (A3) and the definition of α that

partL(ααminus αΛ)

=

983133 1

0

983063983133 1

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 1

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983064d[α(v)minus α(v)]

=

983133 1

0

983077983133 12

v

(uprime(α(x))f(x)minus κF (x) + Λ(x))dx+

983133 12

v

([xminus α(x)]d[Λ(x)minus κF (x)])

983078dα(v)

Since α is increasing (A4) and (A5) imply that partL(αα minus αΛ) le 0 Since the optimal actionrule chooses action 1 for type 1 we conclude that α is optimal

Lastly we consider optimality of interval delegation

Proof of sufficiency part of Proposition 3 The induced action function is α(v) = 0 for v lt

clowast2 α(v) = clowast for clowast2 le v le clowast and α(v) = v for v gt clowast

We propose the following multiplier

Λ(v) =

983099983105983105983105983103

983105983105983105983101

κF (clowast2)minus uprime(clowast)F (clowast)minusF (clowast2)clowastminusclowast2 if v lt clowast

κF (v)minus uprime(v)f(v) if clowast le v lt 1

κF (1) if v = 1

Λ is constant on [0 clowast) and it follows from Proposition 3rsquos condition (i) that Λ is increasing on(clowast 1] To see that Λ is increasing at clowast note that condition (ii) holds as an equality for t = clowast

and hence the derivative of the LHS of condition (ii) with respect to t must be negative att = clowast which yields

minusκF (clowast)minus F (clowast2)

clowast2+ uprime(clowast)

f(clowast)clowast2minus (F (clowast)minus F (clowast2))

(clowast2)2le 0

Hence Λ is increasing at clowast It is thus sufficient to show partL(ααminus αΛ) le 0 for all α isin A

33

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

Note that for v isin [clowast 1] v minus α(v) = 0 and the definition of Λ implies that for v isin [clowast 1)

uprime(α(v))f(v)minus κF (v) + Λ(v) = 0

Therefore

partL(ααminus αΛ) =

983133 clowast

0

983061983133 clowast

v

uprime(α(x))f(x)minus κF (x) + Λ(x)ds

983062d[α(v)minus α(v)]

+

983133 clowast

0

983061983133 clowast

v

[xminus α(x)]d[Λ(x)minus κF (x)]

983062d[α(v)minus α(v)]

Since α is constant on [0 clowast2) and [clowast2 clowast] αminusα is increasing on [0 clowast2) and [clowast2 clowast] Hencethe following conditions are sufficient for α to maximize the Lagrangian

983133 clowast

t

983043uprime(clowast)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)] le 0

for t isin [clowast2 clowast] with equality at t = clowast2 and

983133 clowast2

s

983043uprime(0)f(v)minus [κF (v)minus Λ(v)]

983044dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)] le 0

for s isin [0 clowast2)

Note that983125 clowast

t

983043F (v) + (v minus clowast)f(v)

983044dv = (clowast minus t)F (t) and Λ is constant on [0 clowast) Hence

using the definition of Λ we get that for t isin [clowast2 clowast]

983133 clowast

t

(uprime(clowast)f(v)minus [κF (v)minus Λ(v)]) dx+

983133 clowast

t

(v minus clowast)d[Λ(v)minus κF (v)]

=

983133 clowast

t

983061uprime(clowast)f(v)minus

983063κF (v)minus κF (clowast2) +

1

clowast minus clowast2

983133 clowast

clowast2

uprime(clowast)dF (x)

983064983062dv minus κ

983133 clowast

t

(v minus clowast)dF (v)

=uprime(clowast)[F (clowast)minus F (t)]minus uprime(clowast)[F (clowast)minus F (clowast2)]clowast minus t

clowast minus clowast2+ κ(clowast minus t)[F (clowast2)minus F (t)]

=minus [uprime(clowast) + κ(clowast minus t)][F (t)minus F (clowast2)] + (tminus clowast2)uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2

le0

where the inequality is by Proposition 3rsquos condition (ii) and holds with equality for t = clowast2

Analogously note that983125 clowast2

sF (v) + vf(v)dv = clowast2F (clowast2) minus sF (s) and Λ is constant on

34

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

[0 clowast2] Hence for s isin [0 clowast2]

983133 clowast2

s

(uprime(0)f(v)minus [κF (v)minus Λ(v)]) dv +

983133 clowast2

s

vd[Λ(v)minus κF (v)]

=uprime(0)[F (clowast2)minus F (s)] + Λ(s)(clowast2minus s)minus κ[clowast2F (clowast2)minus sF (s)]

=[uprime(0)minus κs][F (clowast2)minus F (s)]minus uprime(clowast)F (clowast)minus F (clowast2)

clowast minus clowast2(clowast2minus s)

le0

where the inequality is by Proposition 3rsquos condition (iii) We conclude that α is optimal

A2 Necessary Conditions

Lemma A4 Suppose Condition LQ holds If αlowast is deterministic and solves problem (P) then it alsosolves problem (R)

Proof The proof is by contraposition assuming there exists α isin A that is feasible for (R) andachieves a strictly higher objective value in (R) than αlowast we will construct a solution to (P) thatachieves a strictly higher objective value than αlowast

Claim 1 There exists α isin A that is feasible for (R) satisfies α(v) le 1 for all v vα(v) minusα(v)2

2minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 and achieves a weakly higher objective value

in problem (R) than α

We can assume α(v) le 1 since u is decreasing above 1 Now suppose instead that vα(v)minusα(v)2

2minus

983125 v

0α(s)ds gt 0 for some v such that α(v) = 1 Consider an auxiliary setting in which

a principal chooses a pair of functions (α t) and an agent with type v gets utility vα(v) minusα(v)2

2minus t(v) Since α is monotonic it follows from standard arguments that there exist transfers

t [0 1] rarr R such that (α t) is incentive compatible in the auxiliary setting (eg Amador andBagwell 2013) For all v these transfers satisfy t(v) minus t(0) = vα(v) minus α(v)2

2minus

983125 v

0α(s)ds ge 0

where the inequality holds because α is feasible for (R) Define (α t) by setting (α(v) t(v)) =

(α(v) t(v)) or (α(v) t(v)) = (1 t(0)) whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent) Note that t(0) = t(0) which togetherwith t(v) ge t(0) implies t(v)minus t(0) ge 0 with equality for any v such that α(v) = 1

Observe that (α t) corresponds to an incentive compatible direct mechanism indeed iftype v strictly prefers (α(vprime) t(vprime)) to (α(v) t(v)) then v also strictly prefers (α(vprime) t(vprime)) to(α(v) t(v)) contradicting the assumption that (α t) is incentive compatible It follows from

35

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

the standard characterization of incentive compatible mechanisms that α is increasing

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds = t(v)minus t(0) ge 0

with the inequality holding as equality for v such that α(v) = 1

Finally note that α(v) le α(v) le 1 for all v Also t(v)minus t(0) le t(v)minus t(0) which implies

vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds le vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

It follows that α achieves a weakly higher objective value in problem (R) ⋄

Claim 2 Let α isin A be feasible for (R) and satisfy α(v) le 1 and vα(v) minus α(v)22 minus983125 v

0α(s)ds = 0 for all v such that α(v) = 1 There is a stochastic mechanism m such that

for all v Probm(v)(a le 1) = 1 Em(v)[a] = α(v) and Em(v)

983147vaminus a2

2

983148minus

983125 v

0Em(s)[a]ds = 0

Intuitively this is because Vetoerrsquos utility function is quadratic and we can use noise as asubstitute for transfers We provide an explicit construction of the mechanism m below

For any v such that α(v) = 1 define m(v) to put mass 1 on action 1 Now fix arbitrary v

such that α(v) lt 1 and arbitrary d isin (minusinfin 0] and let t1(d) =1minusα(v)1minusd

Then t1(d)d+(1minus t1(d))1 =

α(v) for all d Moreover for any real number r we can choose d isin (minusinfin 0] small enough suchthat

minus1minus α(v)

1minus dd2 minus

9830611minus 1minus α(v)

1minus d

983062le r

because the LHS rarr minusinfin as d rarr minusinfin Hence by choosing d small enough we get

vα(v)minus t1(d)d2

2minus (1minus t1(d))

1

2minus

983133 v

0

α(s)ds le 0

Given v isin [0 1] and t2 isin [0 1] we define m(v) to put probability t2 on action α(v) probability(1 minus t2)t1(d) on action d and probability (1 minus t2)(1 minus t1(d)) on action 1 It follows from theabove that m(v) satisfies Em(v)[a] = α(v) Probm(v)(a le 1) = 1 and we can choose t2 isin [0 1]

such thatEm(v)

983063vaminus a2

2

983064minus

983133 v

0

Em(s)[a]ds = 0

Defining m(v) in this way for all v such that α(v) lt 1 the claim follows ⋄

36

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

We conclude that m is feasible for (P) Therefore

983133 1

0

u(αlowast(v))dF (v) =

983133 1

0

983061u(αlowast(v))minus κ

983063vαlowast(v)minus αlowast(v)2

2minus

983133 v

0

αlowast(s)ds

983064983062dF (v)

lt

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

le983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v) (A6)

where the equality holds because αlowast is feasible for (P) the first inequality holds because weassume that α achieves a strictly higher value than αlowast and the second inequality holds byClaim 1

Under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γ Hence for any a b le 1 and λ isin [0 1]some algebra shows that

u (λa+ (1minus λ)b) + κ[λa+ (1minus λ)b]2

2= λ

983063u(a) + κ

a2

2

983064+ (1minus λ)

983063u(b) + κ

b2

2

983064

Since Probm(v)(a le 1) = 1 and Em(v)[a] = α(v) for all v expression (A6) therefore equals

983133 1

0

983061Em(v)

983063u(a)minus κ

983061vaminus a2

2

983062983064+ κ

983133 v

0

Em(s)[a]ds

983062dF (v)

Since m is feasible for (P) this expression equals983125 1

0Em(v)[u(a)]dF (v) This contradicts the

assumption that αlowast solves (P) and we conclude that αlowast solves (R)

Let φ denote the objective function in (R) The set of feasible solutions for (R) is convexand optimality of α therefore implies partφ(αα minus α) le 0 for any α isin A that is feasible for (R)Recall the assumption F (1) = 1

φ(α) =

983133 1

0

983061u(α(v))minus κ

983063vα(v)minus α(v)2

2minus

983133 v

0

α(s)ds

983064983062dF (v)

partφ(ααminus α) =

983133 1

0

983061[uprime(α(v))minus κ[v minus α(v)]](α(v)minus α(v)) + κ

983133 v

0

α(s)minus α(s)ds

983062dF (v)

=

983133 1

0

983063uprime(α(v))minus κ

983063v minus α(v)minus 1minus F (v)

f(v)

983064983064(α(v)minus α(v))dF (v) (A7)

Lemma A5 Suppose Condition LQ holds If a delegation set containing the interval [a b] sube [0 1] isoptimal then κF (v)minus uprime(v)f(v) is increasing on [a b]

37

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

Proof Suppose a delegation set containing the interval [a b] is optimal and let α denote thecorresponding allocation rule Suppose to the contrary that κF (v) minus uprime(v)f(v) is strictly de-creasing for some v isin [a b] Then uprime(v)f(v) minus κf(v)[minus1minusF (v)

f(v)] is strictly increasing on some

interval [d e] sub [a b] with d ∕= e that contains v

Set α(v) = α(v) for v isin [d e] and α(v) = d for v isin [d e] Then α isin A and it is feasible for (R)Since α(v) = v for v isin [d e] it follows from (A7) that partφ(ααminus α) gt 0 a contradiction

Proof of the necessity part of Proposition 1 Suppose Condition LQ holds The result fol-lows directly from Lemma A5

Proof of the necessity part of Proposition 2 Suppose Condition LQ holds and let α isin A bethe action rule induced by the delegation set 0 1 Fix s isin [0 12) and t isin (12 1] ε isin (0 1)

and define

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if v isin [0 s)

ε if v isin [s 12)

1minus 12minusstminus12

ε if v isin [12 t)

1 if v isin [t 1]

Note that

983133 12

s

983063v minus 1minus F (v)

f(v)

983064dF (θ) = s[1minus F (s)]minus 12[1minus F (12)] and

983133 t

12

983063v minus 1minus 1minus F (v)

f(v)

983064dF (θ) = F (t)(tminus 1)minus F (12)(12minus 1)minus t+ 12

It follows that

partφ(ααε minus α) =ε

983133 12

s

983061uprime(0)minus κ

983063v minus 1minus F (v)

f(v)

983064983062dF (v)

minus 12minus s

tminus 12ε

983133 t

12

983061uprime(1)minus κ

983063v minus 1minus 1minus F (v)

f(v)

983064983062dF (v)

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus s+ κ[1minus F (12)]

983062

minus ε12minus s

tminus 12[uprime(1) + κ(1minus t)] (F (t)minus F (12) + κ[(12minus t)F (12) + tminus 12])

=ε(12minus s)

983061[uprime(0)minus κs]

F (12)minus F (s)

12minus sminus [uprime(1) + κ(1minus t)]

F (t)minus F (12)

tminus 12

983062

38

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

Therefore partφ(ααε minus α) le 0 for all ε gt 0 s isin [0 12) and t isin (12 1] if and only if thecondition in Proposition 2 holds

Proof of necessity part of Proposition 3 Suppose Condition LQ holds and clowast isin (0 1) Letα isin A be the action rule induced by the delegation set [clowast 1] We prove necessity of eachcondition in Proposition 3 in order

Condition (i) This follows from Lemma A5

Condition (ii) Fix t isin (clowast2 clowast) and ε gt 0 Let a(ε) be the positive solution to (clowast minus t)a +

a22 = ε(tminus clowast2) and define αε by

αε(v) =

983099983105983105983105983103

983105983105983105983101

α(v) if v isin [clowast2 clowast + a(ε)]

clowast minus ε if v isin [clowast2 t)

clowast + a(ε) if v isin [t clowast + a(ε)]

Note that for any ε gt 0 small enough (so that clowast minus ε gt clowast2 and clowast + a(ε) lt 1) αε isin A Bydefinition of a(ε) for any v isin [0 1] vαε(v)minus [αε(v)]2

2minus

983125 v

0αε(s)ds ge 0 and hence αε is feasible

for (R) Therefore if α is optimal partφ(ααε minus α) le 0

Note that

partφ(ααε minus α) =minus ε

983133 t

clowast2

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+ a(ε)

983133 clowast

t

983061uprime(α(v))f(v)minus κf(v)

983063v minus clowast minus 1minus F (v)

f(v)

983064983062dv

+

983133 clowast+a(ε)

clowast(clowast + a(ε)minus v)

983061uprime(α(v))f(v)minus κf(v)

983063minus1minus F (v)

f(v)

983064983062dv

By the implicit function theorem limεrarr0a(ε)ε

= tminusclowast2clowastminust

It follows that the last integral is of ordero(ε) Also integration by parts implies that for x y isin R

983125 y

x(f(v)(v minus clowast) minus [1 minus F (v)])dv =

F (y)(y minus clowast)minus F (x)(xminus clowast)minus y + x We conclude

limεrarr0+

partφ(ααε minus α)

ε=minus uprime(clowast)[F (t)minus F (clowast2)] + κ[F (t)(tminus clowast) + F (clowast2)(clowast minus clowast2)minus t+ clowast2]

+tminus clowast2

clowast minus t[uprime(clowast)(F (clowast)minus F (t))minus κ(clowast minus t)(F (t)minus 1)]

=minus clowast minus clowast2

clowast minus tuprime(clowast)[F (t)minus F (clowast2)] + κ(clowast minus clowast2)[F (clowast2)minus F (t)]

39

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

+tminus clowast2

clowast minus tuprime(clowast)(F (clowast)minus F (clowast2))

=(clowast minus clowast2)(tminus clowast2)

clowast minus t

times983069minus[uprime(clowast) + κ(clowast minus t)]

F (t)minus F (clowast2)

tminus clowast2+ uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070

Since partφ(ααε minus α) le 0 for all ε gt 0 the last expression is negative for all t isin (clowast2 clowast) whichimplies condition (ii)

Condition (iii) Fix s isin [0 clowast2) and ε gt 0 Let

αε(v) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 for v lt s

ε if v isin [s clowast2)

clowast minus a(ε) if v isin [clowast2 clowast minus a(ε))

v if v ge clowast minus a(ε)

where a(ε) ge 0 satisfies

clowast2(clowast minus a(ε))minus (clowast minus a(ε))22minus (clowast2minus s)ε = 0

lArrrArr (clowast minus clowast2)a(ε)minus (a(ε))2 = (clowast2minus s)ε

For ε small enough there is a real solution a(ε) ge 0 and a simple calculation shows that αε isfeasible for (R) Also note that limεrarr0

a(ε)ε

= clowast2minussclowastminusclowast2

It follows from (A7) that

partφ(ααε minus α) =ε

983077983133 clowast2

s

uprime(α(v))minus κ

983063v minus 1minus F (v)

f(v)

983064dF (v)

983078

minus a(ε)

983063983133 clowast

clowast2

uprime(α(v))minus κ

983063v minus clowast minus 1minus F (v)

f(v)

983064dF (v)

983064+ o(ε)

Using integration by parts we conclude

limεrarr0+

partφ(ααε minus α)

ε= uprime(0)[F (clowast2)minus F (s)]minus κ[clowast2F (clowast2)minus sF (s)minus clowast2 + s]

minus clowast2minus s

clowast minus clowast2[uprime(clowast)[F (clowast)minus F (clowast2)] + κ(clowast minus clowast2)[1minus F (clowast2)]]

= (clowast2minus s)

983069[uprime(0)minus κs]

F (clowast2)minus F (s)

clowast2minus sminus uprime(clowast)

F (clowast)minus F (clowast2)

clowast minus clowast2

983070le 0

40

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

which yields condition (iii)

B Proofs of Corollaries 1 2 and 3

B1 Proof of Corollary 1

Since u is concave uprime is decreasing on [0 1] Recall κ ge 0 Hence if the type density f isdecreasing on [0 1] then κF minusuprimef is increasing on [0 1] The result follows from Proposition 1

B2 Proof of Corollary 2

As κF (v) minus uprime(v)f(v) is continuous on [0 1] it is increasing on [0 1] if its derivative ispositive for all v isin [0 1) The derivative is (κ minus uprimeprime(v))f(v) minus uprime(v)f prime(v) which is larger thanuprimeprime(v)f(v)minus uprime(v)f prime(v) The latter function is positive for all v isin [0 1) if

infvisin[01)

minusuprimeprime(v)

uprime(v)ge sup

visin[01)

f prime(v)

f(v)

The RHS above is finite since f is continuously differentiable and strictly positive on [0 1]Therefore κF (v) minus uprime(v)f(v) is increasing on [0 1] when the LHS above is sufficiently largeThe result follows from Proposition 1

B3 Proof of Corollary 3

Assume Condition LQ We prove the result by establishing that (i) logconcavity of f on[0 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied and (ii)if γ gt 0 (equivalently given Condition LQ u is strictly concave) or f is strictly logconcave on[0 1] then among interval delegation sets there is a unique optimum

As introduced in Section 4 Proposerrsquos expected utility from delegating the interval [c 1]with c isin [0 1] is

W (c) equiv u(0)F (c2) + u(c)(F (c)minus F (c2)) +

983133 1

c

u(v)f(v)dv (B1)

As shorthand for the function in condition (i) of Proposition 3 define

G(v) = κF (v)minus uprime(v)f(v) (B2)

We establish some properties of the W and G functions

41

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

Lemma B1 Assume Condition LQ and f is logconcave on [0 1] The functions W and G defined by(B1) and (B2) are respectively quasiconcave and quasiconvex on [0 1] both strictly so if either γ gt 0

or f is strictly logconcave on [0 1] Furthermore for any clowast isin argmaxcisin[01] W (c) Gprime(clowast2) le 0 ifclowast gt 0 and Gprime(clowast) ge 0 if clowast lt 1

Proof The proof proceeds in four steps Throughout we restrict attention to the domain [0 1]

for the type density Step 1 shows that G is (strictly) quasiconvex and that v Gprime(v) = 0 isconnected Step 2 shows that W can be expressed in terms of Gprime Step 3 establishes that givenany maximizer clowast of W G is decreasing on [0 clowast2] and increasing on [clowast 1] Step 4 establishesthe (strict) quasiconcavity of W Note that under Condition LQ κ equiv infvisin[01) minusuprimeprime(v) = 2γuprime(v) = 1minus γ + 2γ(1minus v) and hence G(v) = 2γF (v)minus (1minus γ + 2γ(1minus v))f(v)

Step 1 We first establish that G is (strictly) quasiconvex and that v Gprime(v) = 0 is con-nected Logconcavity of f implies that its modes (ie maximizers) are connected and more-over f prime(v) = 0 =rArr v is a mode Denote by Mo the smallest mode Since

Gprime(v) = 4γf(v)minus (1minus γ + 2γ(1minus v))f prime(v) (B3)

it holds that signGprime(v) = sign β(v) where

β(v) = 4γ minus f prime(v)

f(v)(1minus γ + 2γ(1minus v))

On the domain [0Mo) f primef is positive and decreasing by logconcavity Furthermore 1 minusγ + 2γ(1 minus v) is positive and decreasing As the product of positive decreasing functions isdecreasing β is increasing on the domain [0Mo) Since β(v) ge 0 when v ge Mo it follows thatβ is upcrossing (once strictly positive it stays positive) and hence G is quasiconvex

We claim v β(v) = 0 is connected which implies the same about v Gprime(v) = 0 Ifγ = 0 then β(v) = 0 lArrrArr f prime(v) = 0 which is a connected set as noted earlier If γ gt 0 thenthe conclusion follows because β is increasing on [0Mo) β(v) gt 0 for v gt Mo (as f prime(v) le 0)and β is continuous Furthermore analogous observations imply that if either f is strictlylogconcave or γ gt 0 then |v Gprime(v) = 0| le 1 and so G is strictly quasiconvex

Step 2 We now show that

W prime(c) =

983133 c

c2

(v minus c)Gprime(v)dv (B4)

42

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

The derivation is as follows

W prime(c) =(F (c)minus F (c2))(1 + γ minus 2γc)minus c

2f(c2)(1 + γ minus γc)

=(1 + γ minus 2γc)

983063983133 c

c2

f(v)dv minus c

2f(c2)

983064minus γ

c2

2f(c2)

=minus (1 + γ minus 2γc)

983133 c

c2

(v minus c)f prime(v)dv minus γc2

2f(c2)

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983063minus983133 c

c2

(v minus c)2f prime(v)dv minus983059 c2

9830602

f(c2)

983064

=minus983133 c

c2

(v minus c)(1 + γ minus 2γv)f prime(v)dv + 2γ

983133 c

c2

2(v minus c)f(v)dv

=

983133 c

c2

(v minus c)Gprime(v)dv

The first equality above is obtained by differentiating Equation B1 and using uprime(c) = 1 +

γ minus 2γc and u(c)minus u(0) = c(1 + γ minus γc) the third and fifth equalities use integration by partsthe last equality involves substitution from (B3) and the remaining equalities follow fromalgebraic manipulations

Step 3 We now establish that for any clowast isin arg maxcisin[01]

W (c) clowast gt 0 =rArr Gprime(clowast2) le 0 and

clowast lt 1 =rArr Gprime(clowast) ge 0

By Step 1 there exist vlowast and vlowast with 0 le vlowast le vlowast le 1 such that Gprime(v) lt 0 on [0 vlowast) Gprime(v) = 0

on [vlowast vlowast] and Gprime(v) gt 0 on (vlowast 1] By (B4) c isin (0 vlowast) =rArr W prime(c) gt 0 and c2 isin (vlowast 1) =rArr

W prime(c) lt 0 Since clowast is optimal clowast gt 0 =rArr W prime(clowast) ge 0 =rArr clowast2 le vlowast =rArr Gprime(clowast2) le 0Similarly clowast lt 1 =rArr W prime(clowast) le 0 =rArr clowast ge vlowast =rArr Gprime(clowast) ge 0

Step 4 Finally we establish that W is quasiconcave strictly if γ gt 0 or f is strictly logcon-cave For this it is sufficient to establish that if c gt 0 and W prime(c) = 0 then W primeprime(c) le 0 with astrict inequality if γ gt 0 or f is strictly logconcave

Differentiating (B4)

W primeprime(c) =c

4Gprime(c2)minus (G(c)minusG(c2)) (B5)

Integrating by parts

983133 c

c2

[(v minus c)Gprime(v) +G(v)]dv = [(v minus c)G(v)]cc2 =c

2G(c2)

43

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

Now fix any c gt 0 such that W prime(c) = 0 By (B4) and the above integration by parts G(c2) =

(2c)983125 c

c2G(v)dv which because G is quasiconvex by Step 1 implies G(c2) le G(c) with a

strict inequality if γ gt 0 or f strictly logconcave Similarly Gprime(c2) le 0 and hence from (B5)we conclude that W primeprime(c) le 0 with a strict inequality if γ gt 0 or f is strictly logconcave

We build on Lemma B1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3

Proof of Corollary 3 If the interval delegation set [clowast 1] is optimal then clowast must maximizeW (c) defined in (B1) Hence if W is strictly quasiconcavemdashas is the case if γ gt 0 or f isstrictly logconcave on [0 1] by Lemma B1mdashthere can be at most one interval that is optimalSo it suffices to establish that if clowast isin argmaxcisin[01] W (c) then [clowast 1] is optimal

To that end we verify that if clowast = 1 the conditions of Proposition 2 are satisfied andif clowast lt 1 then conditions (i)ndash(iii) of Proposition 3 are satisfied Note that condition (i) isimmediate from Lemma B1 As conditions (ii) and (iii) are vacuous for clowast = 0 we need onlyconsider clowast isin (0 1] For any clowast isin (0 1) conditions (ii) and (iii) are jointly equivalent to

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle uprime(clowast)

F (clowast)minus F (clowast2)

clowast2le (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2

for all s isin [0 clowast2) and t isin (clowast2 clowast] Substituting into the middle expression from the first-order condition W prime(clowast) = 0 (ie setting expression (2) equal to zero and rearranging) yields

(uprime(0)minus κs)F (clowast2)minus F (s)

clowast2minus sle (u(clowast)minusu(0))

f(clowast2)

clowastle (uprime(clowast) + κ(clowast minus t))

F (t)minus F (clowast2)

tminus clowast2(B6)

for all s isin [0 clowast2) and t isin (clowast2 clowast] So if (B6) holds for clowast isin (0 1) then the conditions inProposition 3 are verified On the other hand since the condition in Proposition 2 is equiv-alent to the right-most term in (B6) being larger than the left-most term for all s isin [0 clowast2)

and t isin (clowast2 clowast] when clowast = 1 (B6) holding for clowast = 1 implies the condition in Proposition 2Accordingly we fix a clowast gt 0 and verify the two inequalities of (B6) in turn

First inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γa thefirst inequality of (B6) reduces to

(1 + γ minus 2γs)F (clowast2)minus F (s)

clowast2minus sle (1 + γ minus γclowast)f(clowast2) foralls isin [0 clowast2)

It follows from LrsquoHopitalrsquos rule that the above inequality holds with equality in the limit

44

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

as s rarr clowast2 Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s isin [0 clowast2) For any s isin [0 1] let

D(s) = (1 + γ minus γclowast)(F (clowast2)minus F (s))minus (clowast2minus s)(1 + γ minus 2γs)f(s) (B7)

and observe that

part

parts

983063(1 + γ minus 2γs)

F (clowast2)minus F (s)

clowast2minus s

983064=

1

(clowast2minus s)2D(s)

So it is sufficient to show that for all s isin [0 clowast2) D(s) ge 0 This holds because D(clowast2) = 0

and for all s lt clowast2

Dprime(s) = (clowast2minus s)[4γf(s)minus (1 + γ minus 2γs)f prime(s)] differentiating (B7) and simplifying

= (clowast2minus s)Gprime(s) substituting from (B3) (B8)

le 0 by Lemma B1

Second inequality of (B6) Using uprime(a) = 1 + γ minus 2γa κ = 2γ and u(a)minusu(0)a

= 1 + γ minus γathe second inequality of (B6) reduces to

(1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2forallt isin (clowast2 clowast]

Using LrsquoHopitalrsquos rule for the limit as t rarr clowast2 and the fact that W prime(clowast) ge 0 by optimality ofclowast gt 0 it follows that

limtrarrclowast2

(1 + γ minus 2γt)F (t)minus F (clowast2)

tminus clowast2= (1 + γ minus γclowast)f(clowast2) le (1 + γ minus 2γclowast)

F (clowast)minus F (clowast2)

clowast2

Hence it is sufficient to show that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast]

Note thatpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (clowast2)

tminus clowast2

983064=

1

(tminus clowast2)2D(t)

where D is defined in (B7) and so

signpart

partt

983063(1 + γ minus 2γt)

F (t)minus F (c2)

tminus c2

983064= signD(t)

Since D(clowast2) = 0 it follows that (1 + γ minus 2γt)F (t)minusF (clowast2)tminusclowast2 is quasiconcave for t isin (clowast2 clowast] if

D is quasiconcave D is quasiconcave because as was shown in (B8) Dprime(t) = (clowast2minus t)Gprime(t)

45

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

which is positive then negative on (clowast2 clowast] by the quasiconvexity of G (Lemma B1)

C Proof of Proposition 4

Proof of Proposition 4(i) Let H(a c) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [c 1] That is

H(a c) =

983099983105983105983105983105983105983105983103

983105983105983105983105983105983105983101

0 if a lt 0

F (c2) if 0 le a lt c

F (a) if c le a lt 1

1 if 1 le a

For any 0 le cL lt cH le 1 the difference H(middot cL) minus H(middot cH) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968Theorem 31 on p 21) that

983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0 =rArr983133 1

0

uprime(a) [H(a cL)minusH(a cH)] da ge (gt)0

when u is strictly more risk averse than u Integrating by parts we obtain

983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0 =rArr983133 1

0

u(a) [H(da cL)minusH(da cH)] le (lt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(u) geSSO Clowast(u)

Proof of Proposition 4(ii) Let density f(v) strictly dominate density g(v) in likelihood ratioon the unit interval ie for all 0 le vL lt vH le 1 f(vL)g(vH) lt f(vH)g(vL) Let w(c v) denoteProposerrsquos payoff under the interval delegation set [c 1] when Vetoerrsquos type is v We have

w(c v) =

983099983105983105983105983103

983105983105983105983101

u(0) if v isin [0 c2)

u(c) if v isin (c2 c)

u(v) if v gtisin (c 1]

Consider any 0 le cL lt cH le 1 The difference w(cH middot) minus w(cL middot) is upcrossing once strictlypositive it stays positive It follows from the variation diminishing property (Karlin 1968

46

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

Theorem 31 on p 21) that

983133 1

0

[w(cH v)minus w(cL v)] g(v)dv ge (gt)0 =rArr983133 1

0

[w(cH v)minus w(cL v)] f(v)dv ge (gt)0

A standard monotone comparative statics argument (Milgrom and Shannon 1994) then im-plies that Clowast(f) geSSO Clowast(g)

D Proof of Proposition 5

Let aU and aI denote proposals in some noninfluential and influential cheap-talk equilib-ria respectively (the latter may not exist) It is straightforward that aU gt 0 and if it existsaI isin (0 1) Since Proposition 5rsquos conclusion is trivial for full delegation (clowast = 0) it suffices toestablish that any optimal interval delegation set [clowast 1] with clowast isin (0 1) has clowast lt minaI aU(By convention minaI aU = aU if aI does not exist)

Plainly aU is a noninfluential equilibrium proposal if and only if

aU isin argmaxa

[u(0)F (a2) + u(a)(1minus F (a2))]

and so if aU lt 1 then it solves the first-order condition

2uprime(a) [1minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D1)

Any influential cheap-talk equilibrium outcome can be characterized by a threshold typevI isin (0 1) such that types v lt vI pool on the ldquoveto threatrdquo message and types v gt vI poolon the ldquoacquiescerdquo message Since type vI must be indifferent between sending the two mes-sages and she will accept either proposal from the Proposer it holds that

vI =1 + aI

2

It follows that aI is an influential equilibrium proposal if and only if

aI isin argmaxa

983063u(0)

F (a2)

F ((1 + aI)2)+ u(a)

9830611minus F (a2)

F ((1 + aI)2)

983062983064

The first-order condition is that function (3) in the main text equals zero ie aI isin (0 1) solves

2uprime(a) [F ((1 + a)2)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D2)

47

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

Note that at a = 0 the LHS is strictly positive Hence if the LHS is strictly downcrossingon (0 1) then Equation D2 has at most one solution in that domain if there is a solutionthen Equation D2rsquos LHS is strictly positive (resp strictly negative) to its left (resp right)furthermore it can be verified that the solution then identifies an influential equilibrium Notethat if there is no solution to Equation D2 on (0 1) then there is no influential equilibrium

Turning to optimal interval delegation recall from Section 4 that the threshold is a zero ofthe function (2) ie clowast isin (0 1) solves

2uprime(a) [F (a)minus F (a2)]minus f(a2) [u(a)minus u(0)] = 0 (D3)

If the LHS is strictly downcrossing on (0 1) then on that domain clowast is the unique solutionto Equation D3 and Equation D3rsquos LHS is strictly positive (resp strictly negative) to thesolutionrsquos left (resp right)

For any a isin (0 1) the LHS of Equation D1 is strictly larger than the LHS of Equation D2which in turn is strictly larger than the LHS of Equation D3 If there is no solution in (0 1)

to Equation D2 then its LHS is always strictly positive and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with aU lt 1 and we are done Soassume at least one solution in (0 1) to Equation D2 Let

a2 = infa isin (0 1) Equation D2rsquos LHS le 0

a2 = supa isin (0 1) Equation D2rsquos LHS ge 0

and analogously define a3 and a3 using Equation D3rsquos LHS The aforementioned ordering ofthe equationsrsquo LHS Equation D2rsquos LHS being strictly positive at 0 and continuity combineto imply 0 le a3 lt a2 lt aU and a3 le a2 with a strict inequality if either a2 lt 1 or a3 lt 1Furthermore aI isin [a2 a2] and clowast isin [a3 a3]

If the LHS of Equation D2 is strictly downcrossing on (0 1) then by the properties notedright after Equation D2 aI = a2 = a2 lt 1 and hence clowast lt minaI aU If the LHS ofEquation D3 is strictly downcrossing on (0 1) then by the properties noted right after Equa-tion D3 clowast = a3 = a3 lt 1 and hence clowast lt minaI aU

E Stochastic Mechanisms can be Optimal

Example E1 Suppose Proposer has a linear loss function v = 0 v = 1 and f(v) is strictlyincreasing except on (12minusδ 12+δ) where it is strictly decreasing Assume |f prime(v)| is constant

48

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49

v10 1

2

f

Figure 3 ndash A density under which no compromise is the optimal delegation set whenProposer has a linear loss function but it is worse than some stochastic mechanism

(on [0 1])30 Take δ gt 0 to be small See Figure 3

Recall that if δ were 0 then no compromise (ie the singleton menu 1) would be optimalby Proposition 2 or the discussion preceding it It can be verified that no compromise remainsan optimal delegation set for small δ gt 0 We argue below that Proposer can obtain a strictlyhigher payoff however by adding a stochastic option ℓ that has expected value 12 and ischosen only by types in (12minus δ 12 + δ)

The stochastic option ℓ provides action 1 minus 12p

with probability p and action 1 with proba-bility 1 minus p For any p isin (0 1) this lottery has expected value 12 Moreover when p = 1

2minus4δ

quadratic loss implies that type 12minusδ is indifferent between ℓ and action 0 while type 12+δ

is indifferent between ℓ and 1 Consequently any type in [0 12minus δ) strictly prefers 0 to both ℓ

and 1 any type in (12minusδ 12+δ) strictly prefers ℓ to both 0 and 1 and any type in (12+δ 1]

strictly prefers 1 to both ℓ and 0

Therefore offering the menu ℓ 1 rather than 1 changes the induced expected actionfrom 0 to 12 when v isin (12minus δ 12) and from 1 to 12 when v isin (12 12 + δ) Since f(v) isstrictly decreasing on (12minus δ 12 + δ) Proposer is strictly better off ⋄

30 As it is nondifferentiable at two points this density violates our maintained assumption of continuousdifferentiability But the example could straightforwardly be modified to satisfy that assumption

49