Multilateral Bargaining with Collective ProposalControl∗

Svetlana Kosterina†

Abstract

I study a model of multilateral bargaining in which multiple proposers simultaneouslymake offers on several pies. I identify a novel source of inefficient delay unique tomultilateral bargaining – free-riding among proposers combined with the variabilityof proposal power. I establish that there exist stationary equilibria with delay andcharacterize the equilibrium agreement sets. In the worst equilibrium agents agree ifand only if the proposal power is sufficiently concentrated. I compare the efficiencyconsequences of different voting rules, showing that voting rules requiring approval bygreater majorities lead to more delay in the worst equilibrium. Agents’ payoffs in theworst equilibrium are maximized if the distribution of the proposal power is such thatpower is concentrated to a moderate extent. Finally, I show that if the proposal powercan be divided finely among the agents, then there are distributions of power such thatin the worst equilibrium, as the number of agents grows large, delay consumes almostall surplus.

1 Introduction

Writing legislative proposals is oftentimes a collective endeavor. When several agents

possess the means and the expertise, as well as the desire to develop certain parts of

a draft legislative bill or a resolution, the outcome is legislative proposals sponsored by

multiple representatives. Indeed, the incidence of co-sponsored draft bills is high in legislative

bargaining and bargaining in international organizations (Fowler 2006, Baller 2017: 475).1

∗I am grateful to my advisor, Wolfgang Pesendorfer, for his support in developing this project. I would liketo thank Faruk Gul and German Gieczewski for insightful conversations. For helpful comments I would alsolike to thank Gleason Judd, Andrew Mack and seminar audiences at Harvard, Princeton and the Universityof Southern California.†Svetlana Kosterina is a PhD student at Princeton University; email: svetlana.kosterina@princeton.edu.1To illustrate, almost half of the bills introduced in the US Congress in 1973-2004 were cosponsored

(Fowler 2006). Similarly, 38 percent of the bills introduced in the Chamber of Deputies in Argentina in1983-2002 (Aleman et al. 2009: 98-99), and most bills introduced in the Chilean Chamber of Deputies in1990-2002 (Crisp, Kanthak, and Leijonhufvud 2004: 710) were cosponsored.

1

Likewise, bargaining within the governing coalition in parliamentary democracies requires

reaching agreement on policies developed by multiple ministers who make proposals on the

different portfolios that they control.

At the same time, the political bargaining process is rife with delays and disagreements.

Negotiations related to government formation in Belgium after the 2010 elections famously

lasted 541 days, the longest it ever took to form a new democratic government after an

election. Brexit negotiations, having started in March 2017, failed to conclude by the

two-year deadline of 29 March 2019, with Prime Minister Theresa May negotiating a delay.

The lack of success in the negotiations has precipitated several reshuffles of the cabinet,

changing the identities of the ministers involved in the bargaining.2

The aim of the present paper is to explore the consequences of collective proposal

control for the dynamics of the bargaining process. Can the possibility of shared proposal

control render the bargaining outcomes inefficient, and, if so, what is the extent of the

inefficiency? What are the conditions under which agents are likely to reach agreement and

when are they likely to disagree? If an institutional designer can choose the distribution of

the proposal power, what distribution should she choose to maximize the efficiency of the

bargaining process?

I present a model that has the realistic features of the bargaining involving multiple

issues and different agents having proposal power on these issues and making offers at the

same time. I connect the shared control of the proposal power to delay and inefficiency in

bargaining. The key dynamic that emerges is that when the proposal power is fragmented,

agents wait and no one makes acceptable offers. Only when the control of the agenda is

unified do things get done.

I start by introducing a model of multilateral bargaining with collective proposal

control. A surplus is to be divided among the agents. The surplus consists of several

parts, which I refer to as issues, or pies. In each period, proposers are selected randomly.

An agent can be a proposer on a subset of the pies. After the proposers have been selected,

each proposer makes offers on the subset of the pies that she controls. Following that, all

agents accept or reject the offers on the table. If at least q agents accept, then the game

ends and agents receive their payoffs. If fewer than q agents accept, then agreement is not

reached and in the next period proposers are again drawn randomly.

When evaluating the consequences of bargaining procedures, I consider the whole range

of outcomes and judge the game by its worst equilibrium. I study the worst equilibrium

because robustness is an important consideration in designing political institutions: an

2See, for example, “Theresa May’s new-look cabinet” (2018), https://www.bbc.com/news/

uk-politics-44775197.

2

institutional designer would find it desirable to craft the rules in such a way as to ensure

that the institutions work well no matter which equilibrium agents happen to play.

My first finding is that the model admits stationary equilibria that are inefficient

because they feature a delay in agreement. The main reason for the delay is free-riding

among proposers: each proposer would like the other proposers to share surplus with the

non-proposers to secure their support. Delay occurs because, given the conjecture that

the other proposers are offering zero to all other agents under a particular realization of

the proposal power, the equilibrium value can be large enough that a proposer is unwilling

to make the offers leading to agreement all by herself. Thus the reason for the delay is

twofold: it is the free-riding problem facilitated by the simultaneity of the proposals and the

variability of the proposal power over time that enable the proposers to forego agreement

today in hopes of reaching a better deal tomorrow. Importantly, a one-period version of the

model I consider does not admit inefficient equilibria – the prospect of controlling more pies

in the future is crucial for generating the inefficient delay. The reason why delay happens in

the model is, to the best of the author’s knowledge, novel in the literature.

The possibility of delay in a stationary equilibrium contrasts starkly with the nature

of equilibria in a closely related model of legislative bargaining due to Baron and Ferejohn

(1989), which features only one pie and a single proposer in each period. Whereas in the

Baron-Ferejohn model in any stationary equilibrium agents must agree immediately, the

model with multiple pies and multiple proposers generates very different agreement patterns,

admitting equilibria with delay and allowing for rich characterization results.

My model explains why reaching agreement is particularly difficult in multilateral

bargaining. Because in a bilateral setting my model yields immediate agreement in the unique

equilibrium, the reasons for delay isolated by my model are unique to the multilateral setting.

This is in contrast to the existing models producing delay in bargaining, which feature delay

in both bilateral and multilateral environments.

The second major result of the paper is a characterization of the worst equilibrium.

Towards establishing the result, I introduce a measure of resources available for overcoming

gridlock at each realization of the proposal power. This measure is the resources controlled

by the most powerful proposer divided by the number of supporters needed. I show that a

subset of the proposing partitions is an agreement set in the worst equilibrium if and only

if it is the set of the partitions with sufficiently large gridlock resources. Interpreting higher

gridlock resources as concentration of proposal power, it is the set of the partitions under

which power is sufficiently concentrated.

The third contribution of the paper is a comparison of the efficiency properties of

different voting rules. I show that voting rules which require approval by greater majorities

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lead to greater delay in the worst equilibrium. Thus in the model in the present paper,

supermajority rules are inferior to the majority rules, and the unanimity rule is the worst

among all voting rules.

The fourth contribution is a characterization of the set of all equilibrium agreement sets.

I find that if a set of proposing partitions is an equilibrium agreement set in some equilibrium

and we make the partitions in this set more likely, then this set remains an equilibrium

agreement set. I also show that there is a subset of the proposing partitions, which I call

the guaranteed agreement set, that is a subset of every equilibrium agreement set no matter

what the distribution of the proposal power is. This is a set of the proposing partitions

under which gridlock resources are sufficiently large. The proposing partition under which

one proposer controls all the pies is always in the guaranteed agreement set. Finally, I show

that any superset of the worst equilibrium agreement set is an equilibrium agreement set of

some equilibrium and provide a condition under which the set of all equilibrium agreement

sets is exactly the set of all the supersets of the worst equilibrium agreement set.

The fifth major result of the paper concerns the optimal distributions of the proposal

power. I consider an institutional designer who is tasked with choosing the distribution of the

proposal power to maximize the agents’ payoffs in the worst equilibrium. The institutional

designer faces some constraints on the distributions she can choose: the probabilities she

assigns to each proposing partition have to exceed certain thresholds. I show that the optimal

distribution of the proposal power is one in which power is concentrated to a moderate extent.

That is, the probability mass function of the distribution is single-peaked: it puts most of

the weight on a partition under which the gridlock resources are intermediate and puts as

little weight as possible on the partitions under which the gridlock resources are either very

small or very large.

Interestingly, if the institutional designer was unconstrained in choosing the

distribution of the proposal power, then by giving all proposal power to one individual

she could ensure immediate agreement. Alternatively, giving some proposal power to all

agents whose support is needed would also result in immediate agreement. Thus both

very concentrated and very egalitarian distributions of the proposal power lead to efficient

outcomes, and it is the intermediate cases that are prone to inefficient delay.

I also provide comparative statics on the features of the worst equilibrium as the

distribution of the proposal power changes. If we compare two distributions of the proposal

power, f and f ′, such that f dominates f ′ in the sense of first order stochastic dominance

(which means that f ′ puts greater mass on the proposing partitions with smaller gridlock

resources), then the agreement set in the worst equilibrium is weakly larger under f ′. On

the other hand, the payoff in the worst equilibrium may be strictly smaller under f ′.

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The final contribution of the paper lies in showing that if the proposal power can

be divided very finely, then, provided that the number of agents engaged in bargaining

is sufficiently large, the delay in the worst equilibrium can consume almost all surplus.

This result can rationalize the common intuition that, as the number of agents involved in

decision-making becomes excessively large, the outcomes of the decision-making worsen.

The rest of the paper proceeds as follows. Section 2 discusses the applications of the

model. Section 3 introduces the model. Section 4 defines the equilibrium, provides an

example of delay and introduces useful definitions. Importantly, subsection 4.3 introduces

a measure of resources available for overcoming gridlock at each proposing partition and

explains the sense in which this measure can be viewed as a concentration of power. Section

5 is devoted to characterizing the equilibria and comparing the properties of the voting

rules. In particular, subsection 5.2 provides a geometric characterization of the worst

equilibrium, subsection 5.3 compares the properties of different voting rules, and subsection

5.4 characterizes the set of all agreement sets for the equilibria that exist given a distribution

of the proposal power. Section 6 focuses on comparative statics, optimal distributions of the

proposal power and welfare. More specifically, subsection 6.1 analyzes comparative statics

and describes the optimal distributions of the proposal power, whereas subsection 6.2 shows

that if fine division of the proposal power is possible, then delay in bargaining in large groups

can consume almost all available surplus. Section 7 reviews the related literature.

2 Applications

In this section I discuss the applications of the model, which include bargaining

within the governing coalition and in legislatures in parliamentary democracies, legislative

bargaining in non-parliamentary democracies and bargaining in international organizations.3

2.1 Bargaining within the Governing Coalition and in Legislatures

in Parliamentary Democracies

One prominent application of the model is bargaining within the governing coalition

and bargaining in legislatures in parliamentary democracies. In this institutional setting,

3The model can be applied to other settings in which institutional bargaining happens. For example,the model can be applied to multi-union bargaining or bargaining over community development projects.Institutional bargaining is an especially appropriate setting for the model in the present paper because ittakes substantial time for institutions to make offers and to respond to offers, which justifies our focus ondiscount factors δ bounded away from 1.

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failing to get proposals passed has consequences for the identity of the proposers. Cabinets

that fail to get bills passed often get reshuffled. Moreover, in parliamentary democracies

the survival of the government depends on the continued support of the legislative majority.

Should the legislative majority disapprove of the government, it can initiate a motion of

no confidence, which, if approved, would lead to the dissolution of the parliament and new

elections.

The model in the present paper can be mapped to the bargaining in legislatures in

parliamentary democracies as follows. The proposers are the ministers in the Cabinet of

Ministers. The non-proposers are the legislators who are not ministers. The pies that

each proposer controls are the ministers’ portfolios. Each minister makes a proposal on the

portfolio that he controls. An acceptance of all the proposals by a sufficient number of agents

means that the proposed policies are implemented. A rejection of the proposals by enough

agents means that a motion of no confidence is initiated and approved by the legislature.

This leads to new elections, which result in a different governing coalition. In terms of the

model in the present paper, a new round of bargaining begins and proposers on all pies are

drawn randomly.

The model can also be applied to bargaining within the governing coalition in

parliamentary democracies by supposing that the non-proposers are the legislators who are

members of the parties in the governing coalition but are not ministers, and a rejection of

the proposals by enough agents means that a cabinet is reshuffled, with some ministers fired

and new ministers appointed.

Lending credence to our approach, other studies have modeled coalition termination as

a result of a rejection of offers in bargaining. For instance, Lupia and Strom (1995) present

a one-period model in which coalition termination is caused by an exogenous change in the

parameters. As in the model in the present paper, if the recognized proposers are unwilling

to make acceptable offers, then the government is dissolved and new elections are held.

Multiple proposers who make offers simultaneously, multiple pies and an infinite horizon are

among the aspects in which the model in the present paper differs.

2.2 Bargaining in Legislatures and International Organizations

In applying the model in the present paper to bargaining in legislatures in

non-parliamentary democracies, we can interpret different pies as different aspects of a bill

and different proposers having proposal power on different pies as multiple legislators writing

different parts of the bill. The model is likely to apply to complex legislation that deals with

many issues and entails multiple legislators drafting parts of the bill dealing with the different

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issues, as opposed to simple legislation where one legislator can write a policy on a single

issue herself. For instance, consider a bill that deals with economic and environmental policy.

A Senator from New York writing the economic policy part of the bill and a Senator from

Maine writing the environmental policy part of the bill is an example of shared proposal

control. The requirement that agreement is reached on all the pies at the same time can

then be interpreted as saying that the legislature must vote on the bill as a whole and cannot

approve only certain parts of the bill.

The budget approval process in the US Congress is a particularly persuasive example

of an instance of legislative bargaining with collective proposal control. There, multiple

legislators contribute to writing different parts of the bill, either through being members of

the House or Senate Budget Committees and participating in writing the budget resolutions

directly or through being members of other legislative committees and submitting requests

to the budget committees.

The frequency with which multiple proposals on legislative bills are observed is a

testament to the realism of the model. The incidence of bills sponsored by multiple legislators

is high, both in the US Congress and in the legislative bodies in other countries. Almost

half of all bills introduced in the US Congress in 1973-2004 were cosponsored (Fowler 2006).

In the 108th Congress, the mean number of bills cosponsored by a legislator was 285, and

the mean number of cosponsors per bill was 4 (Fowler 2006: 457). 38 percent of the bills

introduced in the Chamber of Deputies in Argentina in 1983-2002 were co-sponsored, and

the mean number of cosponsors per bill was 2.4 (Aleman et al. 2009: 98-99). Similarly, most

bills introduced in the Chilean Chamber of Deputies in 1990-2002 were cosponsored (Crisp,

Kanthak, and Leijonhufvud 2004: 710).

While sometimes the act of cosponsoring a bill may be motivated by signaling

considerations (Kessler and Krehbiel 1996, Wilson and Young 1997) and may entail only

adding one’s name to the list of cosponsors, it may also entail more substantial involvement

in shaping the bill. In particular, Fowler (2006) distinguishes between active and passive

cosponsorship of a bill, arguing that an active cosponsor helps write the bill, while a passive

cosponsor merely declares her support for the bill.

Finally, the model in the present paper can be applied to bargaining in international

organizations. In the United Nations, for example, member states work together to write

draft resolutions and most resolutions have multiple sponsors. Multiple sponsorship is also

common in the governing institutions of the European Union: 24 percent of the total

number of amendments introduced during the seventh term of the European Parliament

were cosponsored amendments in standing committees (Baller 2017: 475).

Both legislative bargaining and bargaining in international organizations are processes

7

rife with delay and disagreements. To the extent that the present model is able to

obtain delay in a stationary equilibrium while the leading alternative model of institutional

bargaining due to Baron and Ferejohn (1989) is not, the present model is likely to match

the observed reality better.

3 Model

There are n agents and m issues, which I also refer to as pies. Each issue generates one

unit of surplus, so that m is the total surplus at stake in the bargaining game. n > m, so

that the number of agents exceeds the number of pies. A proposing partition is a multiset4

of numbers Q = k1, . . . , kj satisfying k ∈ 1, . . . ,m for all k ∈ Q and∑

k∈Q k = m. I use

Q to denote the set of all proposing partitions.

To understand the definition of a proposing partition, consider an example. Suppose

that there are two pies and three agents. Then the set of all proposing partitions is Q =

2, 1, 1. If the proposing partition is 2, then there is exactly one agent who is a

proposer on both pies, and the two other agents are non-proposers. If the proposing partition

is 1, 1, then there is an agent who is a proposer on exactly one pie, there is another agent

who is a proposer on the other pie, and there is one agent who is a non-proposer. Observe that

a proposing partition tells us whether one agent controls both pies or each pie is controlled

by a different agent but it does not tell us the identities of the proposers.

In every period, proposing partitions are chosen randomly according to a distribution

on the space of the proposing partitions Q with a probability mass function f .5 The proposal

power is independent and identically distributed over periods. Given any realized proposing

partition Q, I assume that for every number of pies k ∈ Q and for every pair of agents i, i′,

the probability that agent i is a proposer on k pies under Q is equal to the probability that

agent i′ is a proposer on k pies under Q.

Each proposer simultaneously makes proposals to all other agents on the pies that she

controls.6 If q ≤ n agents agree to the proposals on all m pies, then the game ends and

the agents receive the payoffs from the offers they accepted. In the event that the offers

are accepted, the proposers get to keep the amount of the pies they control that they have

not offered to anyone. If more than n − q agents reject at least one of the offers, then no

4Unlike a set, a multiset allows for multiple instances of its elements. For instance, 1, 1 is a multisetbut not a set.

5Unless explicitly stated otherwise, I assume that the distribution of the proposal power has full support.6Thus if a proposer controls m pies, then a proposal by this proposer is a vector (x1, . . . , xn) such that

xi ∈ [0,m] for all i = 1, . . . , n and∑ni=1 xi = m, with the interpretation that the proposer offers to give the

amount xi of her pies to each agent i.

8

agreement is reached (on any pies) and in the next period proposers are chosen randomly

again. I assume that q > m, that is, the number of agents whose support is needed for an

agreement exceeds the number of pies.7 The agents have a common discount factor δ ∈ (0, 1).

4 Equilibrium and Preliminary Results

4.1 Equilibrium Definition

I focus on pure strategy symmetric stationary subgame perfect equilibria in which the

offers depend only on the proposing partition Q in the current period and the acceptance

decisions depend only on the current offers and the proposing partition Q in the current

period. Consider first a setting where the voting rule is unanimity. Then a pure strategy

symmetric equilibrium is a collection of mappings consisting of a mapping for each proposer

from the space of the proposing partitions to the space of offers made to every other proposer,

a mapping for each proposer from the space of the proposing partitions to the space of offers

made to every non-proposer, and a mapping for every agent from the space of the proposing

partitions and the offers on the table to [0, 1], the feasible probabilities of accepting the

offers.

When the voting rule is not unanimity, that is, when q < n, I assume that the proposers

have access to a public randomization device. The role of the public randomization device is

to ensure that a (symmetric) pure strategy equilibrium exists. In this setting, the proposers

have to coordinate on the identities of the members of the minimal non-proposing winning

coalition, and a public randomization device aids in this coordination.

If, given a proposing partition Q, there are fewer than q proposers, a minimal

non-proposing winning coalition is a subset of the non-proposers such that the number of the

proposers plus the number of the members of this subset is q. If there are at least q proposers,

a minimal non-proposing winning coalition is an empty set. In a setting where the voting

rule is not unanimity, a pure strategy symmetric equilibrium is a collection of mappings

consisting of a public randomization device that, for each proposing partition, generates a

uniform distribution over the minimal non-proposing winning coalitions, a mapping for each

proposer from the space of the proposing partitions to the space of offers made to every

other proposer, a mapping for each proposer from the space of the proposing partitions to

the space of offers made to every member of the minimal non-proposing winning coalition

7A sufficient condition for this assumption to be satisfied is that n > 2m and q > n2 , that is, it is sufficient

that the number of agents is sufficiently large relative to the number of pies and the voting rule requires thesupport of at least a simple majority of the voters.

9

selected by the public randomization device, and a mapping for every agent from the space

of the proposing partitions and the offers on the table to [0, 1]. To summarize, henceforth

an equilibrium is a pure strategy symmetric stationary subgame perfect equilibrium.

An agreement set for the equilibrium σ is a set of proposing partitions Q(σ) that lead

to agreement. Observe that, because I focus on pure strategy equilibria, Q(σ) is well-defined.

I let Ω(f) denote the set of the agreement sets for the equilibria that exist given that the

distribution of the proposal power is f . I define DQ = maxk ∈ Q as the maximum number

of pies that an agent is a proposer on given that the proposing partition is Q.

For convenience, I impose an innocuous genericity assumption: whenever a proposer

weakly prefers to make strictly positive offers rather than zero offers, the proposer strictly

prefers to make these offers.8

I focus on the equilibria in which only non-proposers receive strictly positive offers.

Lemma 1 in the Appendix shows that it is without loss of generality: for any equilibrium

in which some proposers receive strictly positive offers there exists an equilibrium in which

only non-proposers receive strictly positive offers and the agreement set is the same.

I require that agents use stage-undominated voting strategies (Baron and Kalai 1993).9

It can be shown that this implies that the equilibria that I construct are stage undominated

(Baron and Kalai 1993). I also require that proposers do not make (net) offers that leave

them strictly less than their discounted equilibrium continuation values.10 11 Additionally,

in section B in the Appendix I introduce a more stringent notion of weak dominance,

defining proposal stage undominated equilibria and showing that the equilibria I construct

are proposal stage undominated for a wide range of parameters.

I also focus on the equilibria in which proposers make zero offers under the proposing

partitions Q 6∈ Q(σ).12 Lemma 2 in the Appendix shows that this is, in a sense, without loss

of generality: for any equilibrium in which some proposer makes strictly positive offers under

a proposing partition Q 6∈ Q(σ) there exists an equilibrium σ′ with the same agreement set

in which all proposers make zero offers under the proposing partitions Q 6∈ Q(σ′).13

8I state the assumption informally for the ease of exposition. The assumption can be stated formally asrequiring that certain inequalities hold strictly. The assumption is satisfied for generic values of δ.

9See section B in the Appendix for more details.10Given that agents use stage-undominated voting strategies, such offers must be rejected at the voting

stage. The requirement rules out unreasonable equilibria in which proposers vote against their own offers.11Making offers that satisfy this requirement is always feasible: in particular, each proposer keeping all

the pies she controls for herself satisfies the requirement.12A proposer making zero offers means that a proposer is offering to give zero to all other agents and is

keeping all the pies that she controls for herself.13There is some loss of generality involved in determining which equilibria are proposal stage undominated

(the notion of a proposal stage undominated equilibrium is introduced in section B in the Appendix): for a

10

Finally, I introduce the notion of a simple equilibrium. An equilibrium σ is said to be

simple if whenever Q ∈ Q(σ) a proposer makes a strictly positive offer under Q if and only

if she is a proposer on DQ pies. Thus an equilibrium is simple if the only agents who make

strictly positive offers are the most powerful ones.

4.2 Delay: An Example

The first main result of the paper is that in my model there exist parameters such that

there are stationary equilibria with delay. Here I provide a simple example to illustrate the

result.

Suppose that there are four pies to divide, four agents and approval of all four agents

is needed for agreement. Nature chooses the proposing partition 4 with probability a and

chooses the proposing partition 2, 2 with probability 1− a. We conjecture an equilibrium

in which agreement occurs at 4 but not at 2, 2.

Note that then the probability that the proposing partition 4 and agent i is a proposer

is a4. The value of an agent in this equilibrium satisfies

V =a

4(4− 3δV ) +

(1− a

4

)δV

With probability a4, agent i is a proposer on all 4 pies. Because there is agreement

under this proposing partition and there are three non-proposers, agent i makes offers δV

equal to the discounted equilibrium values to each of the three non-proposers, which yields

a payoff of 4− 3δV to agent i. With probability 1− a4, either the proposing partition is 4

but agent i is not the proposer or the proposing partition is 2, 2, in which case there is

disagreement. In either case, agent i receives his discounted equilibrium value δV . Solving

for V , we obtain V = a1+aδ−δ .

In order to support disagreement under the proposing partition 2, 2, we need that

an agent who is a proposer on only two pies prefers not to make acceptable offers to the

non-proposers provided that the other proposer is making zero offers. The payoff to making

acceptable offers to the non-proposers is 2−2δV , while the payoff to disagreeing is δV . Thus

we require that 2 − 2δV < δV , which is equivalent to δV > 23. Thus an equilibrium with

delay exists if the value in this equilibrium is sufficiently high.

Using the formula for V , we can write this as δ(a + 2) > 2. As δ → 1, the left hand

given set of parameters, it may be the case that an equilibrium in which all proposers make zero offers underthe proposing partitions Q 6∈ Q(σ) is proposal stage dominated but there exists a proposal stage undominatedequilibrium in which some proposer makes strictly positive offers under some proposing partitions Q 6∈ Q(σ).

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side goes to a+ 2, while as a→ 0, the left hand side goes to 2δ. We see that an equilibrium

with delay exists if players are sufficiently patient or if the probability that one agent is a

proposer on all four pies (in which case there is agreement) is sufficiently high. Both of these

conditions increase the agents’ value in the equilibrium with delay.

The intuition for the existence of an equilibrium with delay is as follows. Even though

the size of the surplus is constant in every period and it is feasible to offer to each agent

her discounted value in order to induce the agent to agree, because the proposal power is

dispersed, cooperation among proposers is needed to make the agreement happen. If, for

instance, a proposer conjectures that the other proposer is offering to the non-proposer half

of the discounted value, then the proposer strictly prefers to also offer to the non-proposer

half of the discounted value. If, on the other hand, a proposer conjectures that the other

proposer is making zero offers, then in order to induce agreement the proposer would need to

offer the whole discounted value to the non-proposer. Because the proposer controls only half

of the available pies, the amount left to the proposer after making the acceptable offer to the

non-proposer may be sufficiently small to make the proposer unwilling to make the offer. The

proposer finds the burden of making the acceptable offer too heavy if the discounted value

that she needs to offer is large enough, which explains why we needed that the equilibrium

value be sufficiently large for the equilibrium with delay to exist.

4.3 Measure of Resources Available for Overcoming Gridlock

In order to achieve agreement, proposers have to share some of the surplus with the

agents who have no proposal power. This leads to a free-riding problem among proposers:

each would like the other proposers to share surplus with the supporters while keeping her

own surplus. The proposing partitions Q differ in the extent to which they lead to free-riding

problems. In particular, because overcoming the free-riding problem requires resources, the

free-riding problem decreases as the amount of resources that the most powerful proposer

controls increases.

I let ρQ = n−|Q| denote the number of the non-proposers under a proposing partition

Q. I let sQ = maxq − |Q|, 0 denote the number of non-proposers whose support is needed

for agreement under Q provided that all proposers support the agreement. I refer to sQ as

the size of the minimal non-proposing winning coalition, or supporters.14

14Observe that our assumption that q > m implies that q > |Q| for all Q. If instead we had q < |Q|, thatis, if the number of agents whose support was needed for an agreement was smaller than the number of theproposers, then sQ = 0. It can be shown that in this case in any equilibrium there must be agreement undersuch a proposing partition and that all the proposers must vote in favor of it, implying that agreementswould involve non-minimal winning coalitions.

12

Recall that DQ is the number of issues controlled by the most powerful proposer.

Given that all other proposers support the agreement, sQ = q − |Q| is the number of

supporters needed to achieve agreement. Then RQ denotes the measure of resources available

for overcoming gridlock at Q:

RQ :=DQ

sQ + 1

RQ is the ratio of the amount of resources controlled by the most powerful proposer

to the number of agents who need to get paid to achieve agreement. The number of agents

whose support is needed is sQ + 1 because the most powerful proposer also needs to get

paid. If RQ is large enough, then the most powerful proposer has enough resources to render

preventing delay all by herself worthwhile.

Observe that RQ ≥ mn

if either m = q and each proposer controls one issue, in which

case no supporters are needed, or a single proposer controls all issues, in which case there is

no one to free ride. If RQ ≥ mn

, then there cannot be delay in equilibrium for any values of δ

and f . Moreover, lemma 16 in the Appendix shows that RQ is minimized at Q under which

the maximum number of pies controlled by an agent is small – in particular, it is either one

or two.15

The measure RQ has three interpretations. First, it measures the per-supporter

resources available for overcoming gridlock. In line with this interpretation, going forward,

I am going to refer to RQ as gridlock resources. Second, RQ measures the potential for

free-riding: the smaller RQ is, the more severe the free-riding problem. Third, it measures

the concentration of the proposal power, indicating the extent to which proposal power is

allocated unequally, relative to the number of supporters needed.

I next explain in greater detail the sense in which RQ can be viewed as a measure of the

concentration of proposal power. Given the PMF f of the distribution of the proposal power,

I define a CDF F with respect to the order on the space of the proposing partitions induced

by RQ16 as F (Q) =

∑Q′:RQ′≤RQ f(Q′). This order and, therefore, the CDF F depend on the

voting rule q.17

15Section C in the Appendix also provides an extensive discussion of the intuition behind this result.16For simplicity, I assume that the parameters are such that the order on the space of the proposing

partitions induced by RQ is strict.17An example in the Appendix shows that with 3 pies and 4 agents the order on the proposing partitions

induced by RQ differs depending on the voting rules, so that if q = 4, then R3 > R1,2 > R1,1,1, if q = 3,then R3 = R1,2 = R1,1,1, and if q = 2, then R1,2 > R3 > R1,1,1. Lemma 17 in the Appendixshows that, provided that q > m, if DQ1

≥ DQ2, RQ1

≥ RQ2under q = q1 and q2 ≥ q1, then RQ1

≥ RQ2

under q = q2, that is, if the maximum number of pies someone is a proposer on under one partition is greaterthan this number under the other partition and the gridlock resources are greater for the first partition undera voting rule q1, then this inequality is preserved under any voting rule q2 ≥ q1.

13

I first observe that making proposal power more dispersed in a particular manner

increases the measure RQ. Formally, given Q, we construct Q′ by taking k′′ < k pies from a

bundle k ∈ Q such that k 6= DQ, giving some of them to a proposer that controls k ≥ k pies

and giving some of them to a proposer that controls k ≤ k pies or to a non-proposer. Then

|Q| goes up, which implies that sQ goes down, and DQ goes up, so that RQ goes up.

The above construction shows that sharing control of a bundle of pies other than the

largest one increases RQ. The following observations show that sharing control of the largest

bundle of pies sometimes reduces RQ. First, if, given Q, we construct Q′ by giving some of

the pies controlled by the most powerful proposers to other proposers such that no proposer

controls more than DQ pies, then the largest bundle decreases (DQ′ ≤ DQ) and the number

of the proposers stays the same (|Q′| = |Q|). Because |Q′| = |Q| implies that sQ′ = sQ,

the above implies that RQ ≥ RQ′ . Next, if the number of pies is smaller than the number

of agents whose approval is needed (q ≥ m), exactly one agent controls the largest bundle

under Q, and Q′ is obtained from Q by giving some of the pies from the largest bundle to a

non-proposer, then RQ ≥ RQ′ .18 Thus sharing control of the largest bundle with proposers

always reduces RQ, and sharing control with non-proposers sometimes reduces RQ.19

5 Equilibrium Characterization

5.1 The Value Function

I first show that the value of an agent in a symmetric equilibrium can be written

as a function only of the probability of the proposing partitions under which agents agree.

Suppose that in equilibrium there is agreement under partitions P . Then agreement happens

18This is proven in lemma 5 in the Appendix.19It is important to point out that the focus on dividing control over the largest bundles of pies is key

for interpreting RQ as a measure of concentration of proposal power. Suppose that instead we take twobundles, with k1 and k2 pies, controlled by two distinct proposers under Q and satisfying k1 + k2 < DQ. Weconstruct Q′ by giving control over both bundles to one of the two proposers and making the other agent(who is a proposer under Q) a non-proposer under Q′. Then RQ ≥ RQ′ because the fact that k1 + k2 < DQ

implies that DQ = DQ′ and |Q′| < |Q|, so that sQ′ ≥ sQ. That is, the partition under which control over thesmaller bundle is divided has a higher measure RQ than the partition under which a single agent controlsthe smaller bundle.

14

with probability∑

Q∈P f(Q), and we can compute the value function in this equilibrium as

U(P , f) =m

n

∑Q∈P

f(Q) + δ

(1−

∑Q∈P

f(Q)

)U(P , f)

U(P , f) =mn

∑Q∈P f(Q)

1− δ + δ∑

Q∈P f(Q)

For characterizing the worst equilibrium, cutoff strategies are important. A cutoff

strategy is one where agreement sets have the form

Q(σ) = Q : RQ ≥ b

To compute the value function in a cutoff equilibrium profile, suppose that agreement

occurs under all Q such that RQ ≥ b. Observe that 1− F (b) is the probability that Nature

chooses Q with gridlock resources RQ ≥ b. Agreement thus occurs with probability 1−F (b).

The value function is then given by

V =m

n(1− F (b)) + δF (b)V

V (b) =mn

(1− F (b))

1− δF (b)

Observe that V is a non-increasing step function: the larger the cutoff gridlock resources

RQ, the smaller the agreement set and the probability of agreement, and thus the lower the

payoffs. We can write V as

V (b) =m

n︸︷︷︸per-agent surplus

×

1− (1− δ)(1− F (b))

1− δF (b)︸ ︷︷ ︸proportion of surplus lost due to delay

Because m

nis the value in the equilibrium without delay, we can interpret 1− (1−δ)(1−F (b))

1−δF (b)

as a measure of the efficiency loss associated with delay. The efficiency loss decreases in the

agents’ patience δ and goes to zero as δ → 1.

15

5.2 The Worst Equilibrium

This subsection is devoted to characterizing the equilibrium in which agents obtain the

lowest payoffs due to delay. It is important to study the properties of the worst equilibrium

because there is evidence for ambiguity averse behavior, and agents who are ambiguity averse

and believe that Nature is choosing an equilibrium to minimize their payoff will coordinate

on the worst equilibrium. Moreover, the payoffs in the worst equilibrium are a measure

of the quality of the bargaining protocol against worst case outcomes. An institutional

designer concerned with establishing a protocol under which bargaining is guaranteed to

proceed efficiently would find it useful to know which institutional configurations can lead

to inefficient equilibria and how bad the inefficiency can be.

Fixing a distribution of the proposal power, an equilibrium is said to be the worst if

no other equilibrium that exists given this distribution yields strictly lower payoffs to the

agents.20 Theorem 1 characterizes the worst equilibrium.

Theorem 1. In the worst equilibrium, the agreement set is

Cf = Q : RQ ≥ δV (RQ)

Theorem 1 says that a collection of proposing partitions is an agreement set in the worst

equilibrium if and only if it is the set of all partitions Q such that the gridlock resources

RQ exceed δV (RQ), the discounted value function in a cutoff equilibrium profile. That is,

the agreement set is a set of the partitions under which the proposal power is sufficiently

concentrated. Henceforth, I use Cf to denote the agreement set in the worst equilibrium

given that the distribution of the proposal power is f .

To understand the result, two observations are important. First, observe that delay

happens in the model because the burden of making acceptable offers is shifted to certain

proposers only. The incentive to not make acceptable offers (which leads to delay) is the

greatest if there is only one individual who is tasked with making all the offers. Therefore,

the worst equilibrium is one in which, whenever equilibrium prescribes disagreement under a

proposing partition, each proposer believes that the other proposers are making zero offers.

Second, observe that, because the amount left to the proposer is increasing in the

number of pies she controls, if the proposer on the maximum number of pies is unwilling to

make acceptable offers, then neither are proposers on a smaller number of pies. Moreover,

20Recall that we focus on symmetric equilibria, which implies that all agents obtain the same payoffs.The reason agents’ payoffs may differ across equilibria is that the amount of delay in these equilibria maydiffer.

16

because the amount left to the proposer is decreasing in the number of the non-proposers

she has to buy off, if a proposer on a given number of pies is not willing to make acceptable

offers to the non-proposers under some proposing partition, then, keeping the number of pies

a proposer controls fixed, she is not willing to make acceptable offers to a greater number of

non-proposers.

These two observations imply the following. Suppose that in the worst equilibrium

there is disagreement under some proposing partition Q. Then it must be the case that the

proposer on the maximum number of pies prefers to make zero offers under the conjecture

that all other proposers make zero offers. We claim that this implies that there must be

disagreement under all proposing partitions Q′ satisfying Q′ < Q. The reason is that,

because gridlock resources are smaller under Q′, the proposer either controls fewer pies or

needs to buy off more non-proposers. Thus if she was unwilling to do so given the worst

conjecture about the behavior of the other proposers under Q, then she must be unwilling

to do so given the worst conjecture under Q′.

The following Proposition shows that there is a worst equilibrium in which agents

use particularly simple strategies: only the most powerful proposers make offers to the

non-proposers.

Proposition 1. There exists a worst equilibrium that is simple. Moreover, if for all Q,

f(Q) > 0 implies that |k ∈ Q : k = DQ| = 1, then there exists a unique simple equilibrium.

The first part of Proposition 1 asserts that there exists a worst equilibrium that is

simple. To understand why this is the case, let b∗ be the unique b such that δV (b) ≥ b ≥δV (b′) for b′ ≥ b∗. Observe that b∗ is precisely the point at which a proposer on the maximum

number of pies DQ prefers not to make acceptable offers for all Q with RQ below b∗ under the

conjecture that all other proposers are making zero offers and would make acceptable offers

for all Q with RQ above b∗, also under the conjecture that all other proposers are making

zero offers. In a simple symmetric equilibrium, if there are several agents who are proposers

on DQ pies, then they must all make strictly positive acceptable offers to the non-proposers.

However, if the proposer is willing to make acceptable offers alone, then the proposer is

certainly willing to make acceptable offers when the burden of making the offers is shared.

Therefore, there is a simple worst equilibrium.

The condition that for all Q, f(Q) > 0 implies that |k ∈ Q : k = DQ| = 1 states

that if a proposing partition has a strictly positive probability under the distribution of the

proposal power f , then there is exactly one agent who is a proposer on the maximum number

of pies DQ. This condition can be interpreted as an assumption of No Ties at the Top.

The second part of Proposition 1 says that if the assumption of No Ties at the Top is

17

Figure 1: Worst Equilibrium as An Intersection

RQ(gridlock

resources)

δV (b)

R1,1,1,1 R2,2 R1,1,2 R1,3 R4

45-degree line

agreement set in the worst equilibrium

satisfied, then there exists a unique simple equilibrium. Because under this assumption there

is only one proposer on the maximum number of pies, in a simple equilibrium the proposer

must prefer to make acceptable offers for Q in the agreement set and not to make acceptable

offers for Q in the disagreement set under the conjecture that all other proposers are making

zero offers. As explained above, under this conjecture the proposer on DQ pies is willing to

make acceptable offers if and only if Q is such that RQ ≥ δV (RQ), so the simple equilibrium

is unique and coincides with the worst equilibrium.

An implication of Theorem 1 is that there exists a (stationary) equilibrium with delay

if and only if there is Q with RQ < δV (RQ). If, on the other hand, we have RQ ≥ δV (RQ)

for all Q, then the unique equilibrium involves immediate agreement.

Figure 1 illustrates the geometric interpretation of the worst equilibrium as an

intersection. The figure depicts the gridlock resources for the proposing partitions 1, 1, 1, 1,2, 2, 1, 1, 2, 1, 3 and 4 on the x-axis. For this example, I assume that n = 6, δ = 9

10

and the voting rule is unanimity. The discounted value function in a cutoff equilibrium

profile δV is drawn in red, and the 45-degree line is drawn in blue. The figure shows that

Cf , the agreement set in the worst equilibrium, consists of the proposing partitions 2, 2,1, 1, 2, 1, 3 and 4, which are precisely the partitions with RQ above δV (RQ).21

21The details for this example can be found in Proposition 5 in the Appendix.

18

Proposition 5 in the Appendix shows that the payoff in the worst equilibrium is

U(Cf , f) = 59. On the other hand, the maximum attainable payoff is 6

9. Therefore, the

per-agent efficiency loss in the worst equilibrium in a setting with four pies, six agents and

a discount factor of 910

is 19, which is approximately 16% of the attainable surplus.

5.3 Comparison of the Voting Rules

In this subsection I compare the properties of different voting rules. Proposition 2 says

that in the worst equilibrium delay is longer and payoff is lower under the voting rules that

require a larger majority.

Proposition 2. If σq is a worst equilibrium under the voting rule q and q1 ≥ q2, then∑Q∈Q(σq1 ) f(Q) ≤

∑Q∈Q(σq2 ) f(Q) and U(Q(σq1), f) ≤ U(Q(σq2), f).

The reason that smaller majority requirements q generate smaller delay is that, because

under smaller q proposers need fewer supporters to get their proposals passed, they can keep

a greater share of the pies that they control for themselves after making acceptable offers

to the minimal non-proposing winning coalition. This makes it more difficult to sustain

equilibria with delay in which a proposer is unwilling to make acceptable offers because the

amount of pies left to the proposers after making the offers is too small compared to her

discounted value in the future.

Figure 2 provides a geometric intuition for the result. As we increase the majority

requirements, RQ =DQsQ+1

decreases for all Q because more supporters sQ need to be paid.

On the other hand, V (RQ) does not change because the payoff in a symmetric equilibrium

with a given agreement set does not depend on the voting rule. Then the gridlock resources

that make the marginal most powerful proposer indifferent between making acceptable offers

and not go up. Because the agreement set in the worst equilibrium is the set of all partitions

Q with RQ exceeding the gridlock resources that make the proposer indifferent, the agreement

set in the worst equilibrium shrinks, which increases delay and lowers payoffs.

A comparison of the welfare properties of voting rules in my model with the welfare

properties of voting rules in the models of bargaining with stochastic surplus due to Merlo

and Wilson (1995) and Eraslan and Merlo (2002) is of interest. Eraslan and Merlo (2002)

show that in a setting with one proposer and a stochastic surplus the payoffs under unanimity

are higher than the payoffs under non-unanimous agreement rules because agents agree too

soon under non-unanimous voting rules. That is, in their model, delay is efficient, and

non-unanimity rules lead to an inefficiently small amount of delay. In contrast, in my model,

because the size of the surplus is constant, all delay is inefficient, and rules requiring greater

majorities are worse because they induce greater delay.

19

Figure 2: Comparison of the Voting Rules

RQ(gridlock

resources)

δV (b)

R1,1,1,1 R2,2 R1,1,2 R1,3 R4

45-degree line

agreement set under q2

agreement set under q1 > q2

Thus my model yields the conclusions that are the opposite of those reached by Eraslan

and Merlo (2002): while in their model the unanimity rule is better than non-unanimity rules,

in my model non-unanimity rules lead to higher payoffs (in the worst equilibrium) than the

unanimity rule.

5.4 Characterization of All Equilibrium Agreement Sets

The goal of this subsection is to provide a characterization of all the equilibrium

agreement sets (Theorem 2). Before stating the main result of this subsection, I define

the set G.

Definition 1.

G =Q : RQ ≥ δ

m

n

Definition 1 says that G is the set of all proposing partitions Q such that RQ exceeds

δmn

. G is an interval of the proposing partitions obtained by dividing the set of all proposing

partitions into two intervals and choosing the upper one, which implies that it is a set of the

partitions under which the proposal power is sufficiently concentrated.

Theorem 2 characterizes the set of all the equilibrium agreement sets.

20

Theorem 2 (Characterization of All Equilibrium Agreement Sets).

1. (Increasing Mass on Agreement Set)

(a) Q ∈ Ω(f).

(b) Suppose that P ∈ Ω(f). If∑

Q∈P f(Q) ≥∑

Q∈P f(Q), then P ∈ Ω(f)

.

2. (Guaranteed agreement set) G ⊆ P for P ∈ Ω(f).

Moreover, m ∈ G.

3. (Worst Agreement Set Expands) P ⊆ Q : Cf ⊆ P ⊆ Ω(f).

4. if∑

Q6∈Cf f(Q) < f(Q′) for all Q′ ∈ Cf \ G, then

P ⊆ Q : Cf ⊆ P = Ω(f)

The first part of Theorem 2 characterizes the conditions under which a set of proposing

partitions is an agreement set in some equilibrium. The Theorem says that the equilibrium

with the agreement set Q, the set of all proposing partitions, exists under all distributions f .

Moreover, if P is an equilibrium agreement set under f , then if we increase the probability

of the proposing partitions in P , it remains an equilibrium agreement set.

The intuition for the first part of Theorem 2 is as follows. It can be shown that agents

disagree under a proposing partition Q if the equilibrium value is sufficiently large relative to

the gridlock resources RQ. The reason is that a larger equilibrium value means that, in order

to achieve agreement, a proposer needs to give more to the non-proposers and that the value

the proposer receives in the event of disagreement is larger. Both of these considerations

make a proposer less likely to make acceptable offers.

Because the equilibrium value is determined by the probability of the partitions under

which there is agreement, in order to obtain disagreement under Q, we must put a sufficiently

large mass on the agreement partitions. The reason this condition is sufficient for the

existence of an equilibrium with the conjectured agreement set is that, given any proposing

partition Q, we can construct an equilibrium with agreement under Q by conjecturing that

all proposers contribute appropriately sized shares of the sum of the discounted values of the

non-proposers and showing that a proposer then strictly prefers to contribute her share.

The second part of Theorem 2 says that there exists a proposing partition such that all

partitions with greater gridlock resources than in this partition must be in the agreement set

in any equilibrium no matter what the distribution of the proposal power is. Such proposing

partitions comprise the set G, which I call the guaranteed agreement set. The proposing

partition m under which there is exactly one agent who is a proposer on all the pies is

21

always included in the guaranteed agreement set. The intuition for the result is that if a

proposer controls a sufficiently large number of pies or if the number of non-proposers to

buy off is sufficiently small, then a proposer is willing to induce agreement by giving to the

non-proposers their discounted values no matter how large these values might be.

Because the guaranteed agreement set G is the set of all proposing partitions such

that the gridlock resources RQ exceed δmn

, the guaranteed agreement set becomes smaller

as agents become more patient (as δ is increases) and as the competition for the proposal

power decreases (as mn

increases).

The third part of Theorem 2 says that any collection of the proposing partitions

obtained by adding partitions to the worst equilibrium agreement set Cf is an agreement

set in some equilibrium that exists given that the distribution of the proposal power is f .

The reason for this is as follows. As explained above, given any proposing partition Q, we

can construct an equilibrium with agreement under Q. We then have to check that adding Q

to the agreement set does not affect the agents’ incentives under other proposing partitions.

Because agreement can be incentivized under any proposing partition by appropriately

choosing the conjecture about the behavior of the other proposers, it is enough to show

that the agents’ incentives to disagree are not affected. Recall that agents disagree under

a proposing partition Q if the equilibrium value is sufficiently large relative to Q. Because

adding Q to the agreement set increases the equilibrium value and does not change the

highest partition under which there is disagreement, the agents’ incentives to disagree are

preserved.

The last part of Theorem 2 says that if the mass that f puts on the proposing partitions

outside of the worst equilibrium agreement set Cf is smaller than the mass it puts on any

proposing partition that is in the worst equilibrium agreement set but not in the guaranteed

agreement set G, then the set of the collections of the proposing partitions that are agreement

sets in some equilibria is exactly equal to the set of the collections of the proposing partitions

obtained by adding partitions to Cf . In other words, if the condition is satisfied, then,

provided that the distribution of the proposal power is f , there cannot exist an equilibrium

in which there is disagreement under some proposing partition in the worst equilibrium

agreement set Cf . Because the partitions in Cf but outside of G are the ones under which the

concentration of power is intermediate, this condition can be described as the distribution

of the proposal power putting enough mass “in the middle”.

To understand why, recall that agents disagree under a proposing partition Q if the

equilibrium value is sufficiently large relative to the gridlock resources under Q. Moreover,

the value in the worst equilibrium is small enough that a proposer on the maximum number

of pies is willing to make acceptable offers unilaterally under any partition in the agreement

22

Figure 3: Guaranteed agreement set

RQ(gridlock

resources)

δV (b)

R1,1,1,1 R2,2 R1,1,2 δmnR1,3 R4

45-degree line

agreement set in the worst equilibrium

guaranteed agreement set

set Cf . It can be shown that this implies that having disagreement under any partition Q in

the worst equilibrium agreement set requires strictly increasing the equilibrium value. Thus if

we remove a partition from the worst equilibrium agreement set Cf , we must add sufficiently

many partitions outside of Cf to ensure that the equilibrium value strictly increases. However,

if the mass that f puts on the proposing partitions outside the worst equilibrium agreement

set is smaller than the mass it puts on any proposing partition in the worst equilibrium

agreement set, this cannot be done.

Figure 3 illustrates Theorem 2. We see that the guaranteed agreement set G is contained

in Cf , the agreement set in the worst equilibrium. Moreover, both of these sets have a

threshold structure, in the sense that these are sets of proposing partitions Q for which RQ

exceeds some threshold.

To better understand the content of Theorem 2, consider the following example.22

Suppose that there are four pies to divide and six agents and the voting rule is unanimity.

Then the possible proposing partitions are 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 4. Table 1

lists the proposing partitions and the corresponding values of RQ. We see that the gridlock

22The details for this example can be found in Proposition 5 in the Appendix.

23

Table 1: Example: Proposing Partitions and Gridlock Resources

1, 1, 1, 1 2, 2 1, 1, 2 1, 3 4

RQ13

25

12

35

23

Table 2: Distributions of the Proposal Power

1, 1, 1, 1 2, 2 1, 1, 2 1, 3 4

f(Q) 1624

524

124

124

124

g(Q) 124

2024

472

472

172

resources at the proposing partitions are ordered as

R1,1,1,1 < R2,2 < R1,1,2 < R1,3 < R4

To further develop the numerical example, I set δ = 910

and compute the guaranteed

agreement set G. Because G is the set of all Q satisfying RQ ≥ δmn

, we have G = 1, 3, 4:whenever there is one proposer on all the pies or there is one proposer on three pies and a

different proposer on the remaining pie, there must be agreement in every equilibrium no

matter what the distribution of the proposal power is.

I next consider two distributions of the proposal power with probability mass functions

f and g specified in table 2. It can be shown that Cf = 2, 2, 1, 1, 2, 1, 3, 4 and that

there exists an equilibrium with the agreement set P = 1, 1, 1, 1, 1, 1, 2, 1, 3, 4.Observe that 2, 2 is in Cf but not in P . This shows that if the condition

∑Q 6∈Cf f(Q) <

f(Q′) for all Q′ ∈ Cf \G in part 4 of Theorem 2 is not satisfied, then the set of all equilibrium

agreement sets need not coincide with the set of all supersets of the worst equilibrium

agreement set. That is, there may exist equilibria in which there is disagreement under some

partition in Cf .

Note that∑

Q6∈Cg g(Q) = g(1, 1, 1, 1) = 124< 4

72, which implies that the condition in

part 4 of Theorem 2 requiring that∑

Q6∈Cg g(Q) < g(Q′) for all Q′ ∈ Cg \ G is satisfied. Then

part 4 of Theorem 2 implies that Q, Cg = P ⊆ Q : Cg ⊆ P = Ω(g). It can be shown that

Cg = 2, 2, 1, 1, 2, 1, 3, 4. This implies that in any equilibrium that exists given g,

either the agents agree immediately or they disagree if and only if the proposing partition is

24

1, 1, 1, 1.

6 Comparative Statics, Optimal Proposal Power and

Welfare

6.1 Comparative Statics and Optimal Distributions of Proposal

Power

In this subsection I ask the following question: how do the agents’ payoffs in the worst

equilibrium change as the parameters change? I first compare the worst equilibria under two

distributions of proposal power one of which first order stochastically dominates the other,

as well as under different discount factors and different per-agent surplus (Proposition 3)

and then characterize the optimal distribution of the proposal power subject to constraints

(Theorem 3).

The first part of Proposition 3 considers two distributions of proposal power, f and f ,

with f putting greater mass on the proposing partitions under which the proposal power is

more dispersed. The first result of Proposition 3 says that the worst equilibrium agreement

set is weakly larger under f than it is under f . That is, the set of the partitions under

which there is agreement in the worst equilibrium is larger under the distribution that puts

more mass on the partitions with smaller resources for overcoming gridlock. The reason why

this happens is the intersection characterization of the worst equilibrium agreement set: the

worst equilibrium agreement set is the set of all proposing partitions with RQ above the

intersection of δV and the 45-degree line. Because δV decreases as F increases, if we shift

F up, then the point of the intersection decreases, which implies that the worst equilibrium

agreement set expands.

The second result of Proposition 3 says that if the distributions f and f are not too

different, then the agents’ payoffs in the worst equilibrium are lower under f than they

are under f . Thus even though the set of partitions under which there is agreement in

the worst equilibrium is larger under f , the payoffs are smaller under f . The reason for

this is that if we shift sufficiently little mass to the partitions under which there are less

gridlock resources, then the set of the partitions under which there is agreement in the worst

equilibrium does not change. On the other hand, the mass put on the partitions under which

there is agreement in the worst equilibrium decreases, leading to a decrease in the payoff.

The second part of Proposition 3 says that, given the worst equilibrium for some

parameters δ, m and n, if we lower the discount factor δ or the per-agent surplus mn

by a

25

sufficiently small amount, then the payoff in the worst equilibrium decreases. The reason for

this is that, if the change in the parameters is sufficiently small, the set of the partitions under

which there is agreement in the worst equilibrium does not change. On the other hand, more

impatient agents suffer more from delayed agreement, and the equilibrium payoff is lower if

the per-agent share of surplus mn

is lower.

Proposition 3. The following comparative statics hold:

1. Suppose that F dominates F in the sense of FOSD. Then

(a) Cf ⊆ Cf

(b) there is ε > 0 such that if maxQ

(F (Q)− F (Q)

)< ε, then the payoff in the worst

equilibrium is higher under f than it is under f .

2. For all δ ∈ (0, 1) and m, n, there exist δ and M > 0 such that if δ ∈(δ − δ, δ

)and

mn∈(mn−M, m

n

), then the payoff in the worst equilibrium

(a) is higher under δ than under δ and

(b) is higher under m and n than under m and n.

To understand why equilibrium payoffs can decrease as the distribution puts more

mass on the partitions with smaller gridlock resources, recall that in the worst equilibrium

we agree only if the proposing partition has enough gridlock resources. This implies that

under the new distribution the partitions under which we agree are less likely. If the set of

the partitions under which we agree does not change as we change the distribution, then this

increases the probability of delay and thereby lowers the payoffs.

Figure 4 illustrates Proposition 3. The figure shows two discounted value functions

in a cutoff equilibrium profile, V and V , given that the distribution of the proposal

power is f and f respectively and f dominates f in the sense of FOSD. The agreement

sets in the worst equilibrium given f and f are Cf = 1, 1, 2, 1, 3, 4 and Cf =

2, 2, 1, 1, 2, 1, 3, 4, so that Cf ⊆ Cf , as the Proposition claims. Note that the

payoff in the worst equilibrium is given by V evaluated at the first partition after the point

at which δV intersects the 45-degree line. Because V is below V , it intersects the 45-degree

line sooner. Because V is decreasing, this implies that the payoff in the worst equilibrium is

lower under f .

In order to characterize the optimal distribution of the proposal power, I need several

definitions. The first definition deals with the constraints on the distributions of the proposal

power. I assume that there are two CDFs, F and F , satisfying F (Q) ≤ F (Q) for all Q such

that any CDF that is between these two is a CDF of a feasible distribution of the proposal

26

Figure 4: Comparative Statics

RQ(gridlock

resources)

δV (b)

δV (b)

R1,1,1,1 R2,2 R1,1,2 R1,3 R4

45-degree line

agreement set under F

agreement set under F

power. Substantively, my assumption can be interpreted as the designer of an institution

having the ability to choose any distribution of the proposal power such that the mass on

any given proposing partition is neither too large nor too small.

Let F denote the set of all CDFs on Q. The institutional designer can choose any

distribution of the proposal power from the set G(F , F ), which is defined as follows. We

say that G(F , F ) is the feasible proposal power set given the CDFs F and F such that

F (Q) ≥ F (Q) for all Q if

G(F , F

)= F ∈ F : F (Q) ∈

[F (Q), F (Q)

]for all Q

The second key definition I introduce in this section is that of a moderately concentrated

distribution of the proposal power. Given a proposing partition Q and CDFs F1 and F2

satisfying F1(Q′) ≥ F2(Q′) for all Q′, we say that ΦQ,F1,F2 is a moderately concentrated

distribution of proposal power if

ΦQ,F1,F2(Q′) =

F1(Q′) if RQ′ ≥ RQ

F2(Q′) if RQ′ < RQ

Note that the fact that F1(Q′) ≥ F2(Q′) for all Q′ ensures that ΦQ,F1,F2 is a CDF.

27

ΦQ,F1,F2 is the CDF obtained by merging the CDF F2 on the proposing partitions with

gridlock resources below RQ with the CDF F1 on the proposing partitions with gridlock

resources above RQ. I let φQ,F1,F2 denote the PMF associated with the CDF ΦQ,F1,F2 .

The reason that ΦQ,F1,F2 is a moderately concentrated distribution of proposal power is

that there is a proposing partition Q such that the distribution ΦQ,F1,F2 puts as little weight

as possible on the partitions with gridlock resources below RQ and, subject to this constraint,

puts as much weight as possible on Q. The result is a distribution of the proposal power

that puts little weight on the partitions under which the gridlock resources are very small

(which we can interpret as proposal power being very dispersed) and puts little weight on

the partitions under which the gridlock resources are very large (which we can interpret as

proposal power being very concentrated). Instead, the distribution puts most weight on Q,

which is a proposing partition under which the proposal power is concentrated to a moderate

extent.

Given a distribution of the proposal power f , define

Qf = minQ : Q ∈ Cf

Qf is the proposing partition with the smallest gridlock resources among the partitions

in the worst equilibrium agreement set Cf . Observe that we can equivalently write Qf as

Qf = minQ : RQ ≥ δV (RQ).

Theorem 3 states the main result of the section.

Theorem 3 (Optimal Proposal Power). Consider CDFs F and F such that F dominates

F in the sense of FOSD. Then

minP∈Ω

(φQ

f,F ,F

)U(P , φQf ,F ,F

)≥ minP∈Ω(f)

U(P , f)

for all F ∈ G(F , F ).

Theorem 3 says that if Qf is the first proposing partition after the discounted value

function δV crosses the 45-degree line, then φQf ,F ,F, the distribution of the proposal power

concentrated on Qf , maximizes the agents’ payoffs in the worst equilibrium.

Before explaining the intuition for the theorem, it is useful to define a willingness to

disagree function Q 7→ BQ. Given a proposing partition Q such that RQ < mn

, I define BQ

as

BQ =δmn−RQ

δ(mn−RQ

)28

If RQ ≥ mn

, then I let BQ = −∞.

In the denominator of BQ we have the discounted difference between mn

, the expected

payoff conditional on agreement, and the gridlock resources RQ. In the numerator we

have the same difference, but only the per-agent surplus mn

is discounted. Note that BQ

is increasing in δ, so more patient agents are more willing to disagree. BQ is also increasing

is the difference between mn

and RQ, so the smaller the the gridlock resources are as compared

to the expected payoff conditional on agreement, the more the agents are willing to disagree

today and wait for a better draw tomorrow. It can the shown that Cf = Q : F (Q) ≥ BQ,that is, the agreement set in the worst equilibrium is the set of all proposing partitions such

that the CDF F exceeds the willingness to disagree function B.

The intuition for why the optimal distribution of the proposal power is a moderately

concentrated one is as follows. The objective of the institutional designer is to maximize

the payoffs in the worst equilibrium. Take any feasible distribution of the proposal power

f . We know from Theorem 1 that the agreement set in the worst equilibrium is given by an

interval Cf = Q : RQ > RQ′ for some proposing partition Q′.

Consider a constrained maximization problem in which the designer chooses a feasible

distribution of the proposal power f subject to the constraint that the agreement set in the

worst equilibrium does not change. Then the agents’ payoffs are maximized by choosing a

feasible f such that F (Q′) is minimized, since this minimizes delay in the worst equilibrium.

Choosing f such that F (Q) = F (Q) for all Q with RQ ≤ RQ′ and F (Q) = F (Q) for all Q

with RQ > RQ′ accomplishes this goal. Note that, because F (Q) = F (Q) > BQ for all Q

with RQ > RQ′ and F (Q) = F (Q) ≤ F (Q) < BQ for all Q with RQ ≤ RQ′ , the agreement

set in the worst equilibrium does not change. Moreover, because F (Q′) is the lowest value

that F (Q′) can take, F (Q′) is minimized.

Next, we want to choose the optimal worst equilibrium agreement set. Because the

value of an agent in the worst equilibrium strictly decreasing in F (Q′) (where Q′ is defined as

above), we want choose the worst equilibrium agreement set Cf to minimize F (Q′). Because

F is increasing and Cf is an interval consisting of all proposing partitions with sufficiently

large gridlock resources, this amounts to choosing the largest possible Cf . Because, by part

1 of Proposition 3, Cf expands as we choose higher CDFs, this is accomplished by choosing

the highest possible CDF F . Putting the two steps together, we obtain a distribution of the

proposal power that is concentrated on Qf .

Figure 5 illustrates Theorem 3. The figure shows two CDFs, F and F , drawn in red

and blue respectively. The green area between the two CDFs represents the feasible proposal

power set G(F , F ). Qf = 1, 1, 2 is the lowest proposing partition in Cf (according to the

order induced by RQ). The moderately concentrated distribution of proposal power ΦQf ,F ,F

29

Figure 5: Optimal Proposal Power

Q

(proposingpartitionsordered bygridlockresources)

BQ

1, 1, 1, 1 2, 2 1, 1, 2 1, 3 4

1

0

ΦQf ,F ,F(Q)

G(F , F ) F (Q)

F (Q)

Qf

is the CDF drawn in dark green. As can be seen in the figure, ΦQf ,F ,Fis equal to F on the

partitions 1, 1, 1, 1 and 2, 2, which are below Qf , and is equal to F on the partitions

1, 1, 2, 1, 3 and 4, which are greater than Qf .

6.2 Inefficiency in Large Groups

In this subsection I state Theorem 4 which characterizes the efficiency loss associated

with the worst equilibrium when the number of agents is large. The first part of Theorem 4

says that if, as the number of agents converges to infinity, the probability of the proposing

partitions under which an agent controls a non-negligible proportion of surplus becomes

arbitrarily small, then the fraction of the feasible surplus attained in the worst equilibrium

is arbitrarily small. That is, delay consumes almost all surplus.

The second part of Theorem 4 provides a converse to the above statement: if, as

the number of agents converges to infinity, the probability of the partitions under which

a proposer controls a non-negligible proportion of surplus remains substantial, then in any

equilibrium the fraction of surplus attained is bounded away from zero.

Formally, I consider a sequence of games indexed by n, the number of agents. I make

30

the dependence of the parameters and the equilibrium on n explicit by writing mn, sQ,n, DnQ,

fn and σn, and I let Qnγ =

Q :

DnQn≥ γ

.

Theorem 4 (Delay Consumes Almost All Surplus). Suppose that limn→∞mnn

<

limn→∞qnn

. If limn→∞∑

Q∈Qnγfn(Q) = 0 for all γ > 0, then limn→∞

nV σn

mn= 0. If

limn→∞∑

Q∈Qnγfn(Q) > 0 for some γ > 0, then limn→∞

nV σn

mn> 0.

The condition in the first part of the Theorem is not difficult to satisfy. In particular, it

is satisfied if each issue is assigned to every player independently and with equal probability.23

The statement of the Theorem imposes some conditions on the parameters, requiring

that as the number of agents goes to infinity, the number of agents qn needed to reach

agreement is bounded away from the number of pies mn. One way to satisfy the condition

in the first part of the Theorem is to construct a full-support distribution f such that in

the worst equilibrium agents agree under most partitions but the distribution puts most of

the mass on the few partitions under which agents disagree. In this case, because f has full

support, a very fine division of the proposal power is feasible in large groups. That is, it is

feasible for one agent to control a share of the pies that is vanishingly small relative to the

sum of all pies. Then in the worst equilibrium, because agreement is not reached only under

the partitions in which proposers control vanishingly small shares of the sum of the pies,

there exist partitions under which agreement is reached and proposers control vanishingly

small shares of the surplus. Moreover, each agent who is a proposer on this small number

of pies is willing to give acceptable offers to all the non-proposers whose support is needed

even if the other proposers make zero offers.

Because an offer acceptable to a non-proposer must be equal to her discounted value,

this implies that the sum of the discounted values of the non-proposers whose support is

needed is vanishingly small relative to the available surplus. Because the fraction of the

non-proposers whose support is needed is non-negligible relative to the total number of

agents, this implies that the sum of the values of all agents is vanishingly small.

On the other hand, if, as the number of agents converges to infinity, the probability of

the partitions with coarse division of the proposal power does not, then under any partition

resulting in agreement a proposer controls a non-negligible fraction of surplus. If the sum of

the values of the agents was negligible relative to the sum of all pies, this would imply that a

proposer always prefers to make acceptable offers to the non-proposers. This would lead to

agreement under every proposing partition, contradicting our hypothesis that the sum of the

23To see this, observe that if each issue is assigned to every player independently and with equal probability,

then the probability that player i controls αnm issues is(1n

)αnmn(1− 1

n

)(1−αn)mn. If limn→∞ αn > 0, this

probability converges to 0.

31

values of the agents is vanishingly small. Thus it must be the case that in any equilibrium

the sum of the agents’ values is a non-negligible fraction of the feasible surplus.

7 Related Literature

The paper is related to several strands of literature on bargaining with complete

information. Most notably, it is related to the Baron-Ferejohn model of legislative bargaining

(Baron and Ferejohn 1989) in which one proposer on one pie is randomly selected in every

period. This model has become foundational in modeling legislative bargaining and has

been extended in various directions (Eraslan 2002, Kalandrakis 2015, Eraslan and Merlo

2017). Unlike the Baron-Ferejohn model, the model in the present paper allows for multiple

pies and multiple proposers who make the offers on the pies they control simultaneously.

Whereas in the Baron-Ferejohn model there must be immediate agreement in any stationary

equilibrium,24 my model admits stationary equilibria with delay. To the extent that delay is

observed in legislative bargaining and in other settings plausibly modeled within the random

proposer framework, my model can help explain why inefficient delay occurs.

The literature on bargaining with complete information has identified several reasons

leading to delay in agreement: the size of the available surplus changing stochastically25

(Merlo and Wilson 1995, Eraslan and Merlo 2002), non-common prior beliefs about

recognition probabilities26 (Yildiz 2003, Ali 2006), non-stationarity of the equilibrium

(Fernandez and Glazer 1991, Haller and Holden 1990), and repetition of a stage game that

has an inefficient Nash equilibrium (Dekel 1990, Chatterjee and Samuelson 1990).

The reason for delay in the models of bargaining with stochastic surplus is that agents

wait for the size of the surplus to grow. The equilibrium with delay is efficient under the

unanimity voting rule (Merlo and Wilson 1995) but may be inefficient under non-unanimous

voting rules (Eraslan and Merlo 2002). Unlike in the models of bargaining in which the

size of the surplus changes stochastically, the size of the surplus is constant in my model.

Accordingly, the amount of the available surplus is always sufficient to provide all agents

with their discounted equilibrium values. However, because the equilibrium conjectures can

be such that a proposer expects the other proposers to not make positive offers, the burden

24Provided that agents use stage-undominated voting strategies (Baron and Kalai 1993).25Also related is a model by Cai (2000) in which one proposer engages in a sequence of bilateral

negotiations with responders. If a responder accepts an offer, she receives a cash payment immediately andleaves the bargaining process. Cai (2000) shows that these endogenous changes in bargaining environmentcan produce delay.

26Ortner (2013) combines stochastic surplus with non-common prior beliefs about recognitionprobabilities.

32

of providing the non-proposers with their discounted values may be shifted to one proposer

who may control sufficiently few pies to find disagreement in the current period preferable.

Chatterjee and Samuelson (1990) consider an infinite-horizon model of bargaining in

which the stage game is the Nash demand bargaining game. Their model admits one

stationary equilibrium with delay, which involves each agent demanding the entire pie in

every period and never reaching agreement. The key to obtaining delay in this equilibrium

is both the simultaneity of the offers the agents make and the fact that each agent demanding

the entire pie is an equilibrium of the one-period Nash demand bargaining game – that is,

the stage game has an inefficient Nash equilibrium.

The model in the present paper shares with the model of Chatterjee and Samuelson

(1990) the simultaneity of offers as a part of the reason for delay in a stationary equilibrium.

Unlike the model of Chatterjee and Samuelson (1990), the one-period version of my model

does not admit an inefficient Nash equilibrium. Thus the dynamic nature of my model is

crucial for generating delay: in my model agents disagree because sometimes the share of

the pies that a proposer controls is sufficiently small that the proposer prefers to disagree

in the current period in hopes of controlling a larger share of the pies tomorrow. Moreover,

the model in the present paper admits a much richer set of stationarity equilibria with

delay, allows for a geometric characterization of the worst equilibrium and of the set of

all equilibrium agreement sets, connects the extent to which the equilibria are inefficient

to the concentration of the proposal power and permits the identification of the optimal

distributions of the proposal power.

Also related is the literature on multi-issue bargaining (Fershtman 1990, Inderst 2000,

In and Serrano 2004). This literature is concerned with determining the impact of the

bargaining agenda on the outcomes of the bargaining. Only one agent is a proposer in

each period and partial agreements on subsets of the issues are allowed. The papers in this

literature find that agreement is reached without delay in stationary equilibria and that

the bargaining agenda can have distributive effects. In contrast, the model in the present

paper allows for multiple agents making proposals simultaneously, requires that agreement

is reached on all issues at the same time and shows that there exist stationary equilibria in

which agents delay reaching agreement.

Finally, the models in which agents compete for the proposal power (Yildirim 2007,

Ali 2015) are related to the present paper. In those models there is only one proposer at a

time but agents can exert effort to try to become the proposer, whereas in the present model

several agents can be proposers at the same time.

33

Appendix

B Proposal Stage Undominated Equilibria

It is important to point out that the equilibria with delay that I construct do not rely

on the agents using weakly dominated strategies. In order to check whether the equilibria I

characterize are weakly dominated or not, I need to first define the notion of weak dominance

appropriate for my game. A definition of weakly undominated equilibria commonly used in

bargaining games is that of stage undominated equilibria, introduced by Baron and Kalai

(1993). A stage game starts with the proposers being drawn randomly and ends with an

acceptance or a rejection of the offers of the proposers. An equilibrium is said to be stage

undominated if the strategies it induces in every stage game are weakly undominated for any

agent.27 Letting V σ denote the value of an agent in the equilibrium σ, a stage-undominated

voting strategy is a voting strategy such that an agent accepts the offers made to her if and

only if the sum of the offers x satisfies x ≥ δV σ.

It can be shown that, provided that agents use stage-undominated voting strategies,

the equilibria that I construct are both weakly undominated and (generically) stage

undominated. Informally, the reason is that there is no strategy for a proposer that is

optimal no matter what the other agents do: if the other agents accept when the proposer

makes zero offers, then the proposer strictly prefers to make zero offers, while if the other

agents reject when the proposer makes zero offers, then the proposer may strictly prefer to

make strictly positive offers.

One might argue, however, that a more stringent notion of weak dominance is

appropriate for my game. In particular, the reason given in the paragraph above for the

equilibria being stage undominated relies on being able to choose a strategy of the agents at

the point in the game that comes after the offers have been made and it is time to accept or

reject them. This is appropriate for a bargaining game in which there is only one proposer

in every stage game, since in this case there is no way a proposer’s strategy can be weakly

dominated. In my game, on the other hand, because there can be multiple proposers making

offers simultaneously, a more intuitive requirement for a strategy to be weakly dominant is

that it is optimal only no matter what the other proposers do rather than that it is also

optimal no matter what the agents do at a later time when they have to respond to the

offers.

27Recall that a strategy of an agent is weakly undominanted if there does not exist a strategy that weaklydominates it, that is, if there does not exist a strategy that yields to the agent a weakly higher payoff nomatter what strategies the other agents use and a strictly higher payoff for some strategy profile of the otheragents.

34

Motivated by this, I define proposal stage undominated equilibria. I divide each stage

game into a proposal stage game and a response stage game. A proposal stage game starts

with the proposers being randomly drawn and ends with the proposals being made. After

the proposal stage game, a response stage game starts during which all agents accept or

reject the offers on the table.

Given an equilibrium σ, I assign payoffs to each terminal node in a proposal stage game

according to the expected payoff in the subgame that starts after the terminal node of this

proposal stage game, and similarly for a response stage game. An equilibrium σ is said to

be proposal stage undominated if the strategies that it induces in every proposal stage game

and the strategies that it induces in every response stage game are not weakly dominated for

any agent. An equilibrium that is proposal stage undominated is stage undominated, but

the converse need not be true.

Lemma 3 in the Appendix introduces the following condition that is necessary and

sufficient for an equilibrium (in which proposers make zero offers under the proposing

partitions Q 6∈ Q(σ)) to be generically28 proposal stage undominated:

δV σ ≤ m−DQ

q − 1

for all Q 6∈ Q(σ).29

The condition says that the equilibrium value is sufficiently small relative to a quantity

depending on the proposing partitions under which agents disagree in this equilibrium.

The intuition for why this condition is sufficient for the equilibria to be proposal stage

undominated is as follows. For a strategy of proposer i that prescribes that proposer i makes

zero offers under some Q 6∈ Q(σ) to not be weakly dominated by a strategy prescribing that

the proposer makes strictly positive offers, it has to be the case that, given that proposer

i is making zero offers, the other proposers have a strategy that ensures acceptance in the

response stage. For this to be the case, we need that the sum of the pies controlled by

proposers other than i is greater than the sum of the discounted values of the proposers

other than i and the non-proposers. The smaller the equilibrium value is, the easier it is to

satisfy this condition.

28Here genericity refers to the genericity assumption introduced in section 4.1.29That is, pure strategy symmetric stationary subgame perfect equilibria in which agents use

stage-undominated voting strategies and proposers make zero offers under the proposing partitions Q 6∈ Q(σ)are generically proposal stage undominated if and only if this condition holds. If this condition is not satisfied,there may still exist equilibria (that are pure strategy symmetric stationary subgame perfect and withstage-undominated voting strategies) which are proposal stage undominated – and in which some proposersmake strictly positive offers under some proposing partitions Q 6∈ Q(σ).

35

Lemma 4 in the Appendix shows that the following condition is necessary for the

existence of a distribution f such that given f there is a proposal stage undominated

equilibrium with agreement set Q(σ) ⊂ Q:

maxQ 6∈Q(σ)

DQ

sQ + 1< min

Q 6∈Q(σ)

m−DQ

q − 1

Keeping sQ, q and m fixed, the left hand side of the condition is increasing in DQ,

while the right hand side is decreasing in DQ, suggesting that the condition is most likely

to be satisfied when agents disagree only if the maximum number of pies controlled by a

proposer is sufficiently small. Moreover, keeping q, sQ and DQ fixed, the right hand side of

the condition is increasing in m. Then the condition is most likely to be satisfied if m is

close to q.

C Concentration of Proposal Power

This section of the Appendix provides additional discussion that clarifies the manner in

which RQ can be interpreted as a measure of the concentration of proposal power. Towards

this end, the present section explains the intuition behind lemma 13 in the Appendix,

which characterizes the proposing partitions which minimizeDQρQ+1

(lemma 16 shows that,

under the assumption that q > m, the proposing partitions that minimizeDQsQ+1

under some

parameters are the same as the proposing partitions that can minimizeDQρQ+1

– under different

parameters).

In order to state the results of this section, it is helpful to define two special proposing

partitions, D1 and D2. Given m and n, D1 is defined as D1 = Q ∈ Q : DQ = 1. D1 is

the proposing partition in which there is exactly one proposer on each pie. D2 is defined as

D2 = Q ∈ Q : DQ = 2 and |k ∈ Q : k < DQ| ≤ 1. D2 is the proposing partition in which,

if the number of pies m is even, there is exactly one proposer on each pair of pies, and, if

the number of pies is odd, there is exactly one proposer on the m2− 1 pairs of pies and one

proposer on the remaining pie.

Lemma 13 states that if the number of agents is sufficiently small relative to the number

of the pies to be divided, thenDQρQ+1

is minimized under D1. If, on the other hand, the number

of agents is sufficiently large relative to the number of the pies, thenDQρQ+1

is minimized under

D2.

To explain the lemma we use the fact that if a pair of numbers (d, ρ) is the maximal

number of pies DQ an agent is a proposer on under some Q and the number of non-proposers

36

ρQ for the same proposing partition, then (d, ρ) satisfies d ∈ 1, . . . ,m, ρ ∈ N and ρ ≤n − dm

de. Lemma 13 says that arg min(d,ρ):d∈1,...,m,ρ∈N,ρ≤n−dm

de

dρ+1

is given by (2, n − dm2e)

if n + 1 < 2m − dm2e, (1, n − m) if n + 1 > 2m − dm

2e and (2, n − dm

2e), (1, n − m) if

n+ 1 = 2m− dm2e.

Before explaining the intuition for the result, I make several observations highlighting

the significance of the lemma. First, it follows from lemma 6 in the Appendix that in any

equilibrium in which agents disagree under some proposing partitions the equilibrium value

of an agent must exceed the minimal possibleDQρQ+1

. This implies that lemma 13 characterizes

the lower bound of the agents’ equilibrium values. No matter what the distribution of the

proposal power is and no matter which equilibrium the agents play, an equilibrium value of

an agent cannot fall below this bound.

Second, lemmas 8 and 14 in the Appendix imply that if, for instance, m and n are

such thatDD2

ρD2+1

<DD1

ρD1+1

, that is,DQρQ+1

is minimized at D2, then we can find distributions of

proposal power such that in the worst equilibrium agents agree if and only if the proposing

partition is not D2. Thus it is possible that in (the worst) equilibrium agents fail to agree if

each agent controls two pies but reach agreement if each agent controls exactly one pie. On

the other hand, it is not possible that in the worst equilibrium agents fail to agree if each

agent controls strictly more than two pies but reach agreement if each agent controls exactly

one pie.

This provides insight into the manner in whichDQρQ+1

can be interpreted as a measure

of the concentration of proposal power. A proposing partition Q can minimizeDQρQ+1

only if

DQ, the maximum number of pies a proposer can control under this partition, is sufficiently

small. In particular, this can only happen if DQ equals one or two.

We find the minimalDQρQ+1

as follows. Taking DQ as given, Q that minimizesDQρQ+1

should maximize the number of non-proposers ρQ. This is accomplished by requiring, to the

extent that this is feasible, that each proposer is a proposer on the maximal number of pies

DQ. That is, taking the maximum share of pies that a proposer can control as given,DQρQ+1

is minimized when all proposers control the same shares. Having associated the maximal

feasible number of non-proposers with each DQ, we can then choose DQ that minimizes the

resultingDQρQ+1

. It turns out that the resultingDQρQ+1

is minimized at DQ = 1 or DQ = 2.

Equivalently,DQρQ+1

is minimized at D1 or D2.

I next describe the intuition for the result in lemma 13. Increasing the number of pies

that a proposer controls has two effects. On the one hand, it provides greater resources to

each proposer that she can use to make offers to the other agents. Because a proposer with

greater resources has more left over after making acceptable offers to the other agents, this

37

increases the willingness of the proposer to make acceptable offers. On the other hand, it

increases the number of non-proposers whose support is needed for an agreement. Because

each non-proposer in the minimum winning coalition needs to receive an acceptable offer for

an agreement to be reached, a greater number of acceptable offers needs to be made, which

decreases the willingness of the proposer to make acceptable offers. Lemma 13 shows that

if the competition over proposal power is not too intense, then the second effect dominates

andDQρQ+1

is minimized at D2, while if the competition over proposal power is fierce, then the

first effect dominates andDQρQ+1

is minimized at D1.

Interestingly,DQρQ+1

can never be minimized at a partition under which each proposer

controls strictly more than two pies. The reason for this is that the increase in the number

of non-proposers whose support is needed for an agreement is the greatest as we go from one

proposer controlling one pie to one proposer controlling two pies, as compared to going from

one proposer controlling two pies to one proposer controlling a greater number of pies.

D Proofs

Lemma 1 (No Transfers Among Proposers). If there exists an equilibrium in which

under some Q ∈ Q(σ) some proposer receives a strictly positive offer, then there exists an

equilibrium in which no proposer receives strictly positive offers and the set of the proposing

partitions under which there is agreement is the same.

Proof of lemma 1.

Let V denote a value in some equilibrium. We first show that we have δV < 1. Suppose

this was not the case. Then δnV ≥ n. Because n > m, this implies that nV > δnV > m.

But then the sum of the agents’ values nV exceeds the value of the available pies m, which

is a contradiction.

Fix an equilibrium and a proposing partition Q such that there is disagreement under

Q in this equilibrium. Consider the strategy profile in which each proposer makes zero offers

to all other agents and keeps all her pies for herself under Q. Then all non-proposers vote

against the proposals, which implies that there is disagreement under Q, as required.

Next, fix a proposing partition Q such that there is agreement under Q in the

equilibrium that we fixed. Let P denote the set of the agents who are proposers under

Q. For each agent i ∈ P , let Xi denote the sum of the offers he makes to the other agents.

For each agent i, let Yi denote the sum of the offers he receives from the other agents.

We will define a new set of offers as follows. Let xij denote the new offer that agent i

38

makes to agent j. Set xij = 0 for all j ∈ P \ i.

For each i ∈ P , let Zi = Xi − Yi. Zi is the net sum of the offers that proposer i makes

to the other agents. We claim that we must have Zi ≥ 0.

Suppose this was not the case and we had Zi < 0. Fix an agent j such that agent i

receives a strictly positive offer x from agent j (note that the fact that Zi < 0 implies that

such an agent must exist). Consider agent j instead offering x′ < x to agent i (and keeping

x − x′ for herself) and let the resulting net sum of the offers that proposer i makes to the

other agents be denoted by Z ′i. Let x′ < x be chosen such that Z ′i ≤ 0 (note that this is

feasible because Zi < 0).

Making the offer x′ instead of x is a strictly profitable deviation for agent j if agent

i still accepts the offer. Because Z ′i ≤ 0, the payoff to agent i who is a proposer on k ≥ 1

pies from accepting would be at least k ≥ 1, while the payoff to rejecting would be δV .

By the first paragraph of the proof, we have 1 > δV , so that k ≥ 1 > δV . Then agent i

must accept this offer (because agents use weakly undominated voting strategies). This is a

contradiction.

Define

αi =Zi∑j∈P Zj

Note that the fact that there is agreement under Q implies that Zi > 0 for some i ∈ P .

Therefore, we have∑

j∈P Zj > 0, which implies that αi is well-defined.

αi is the ratio of the sum of the net offers proposer i makes to the other agents to the

sum of the net offers all proposers make to the other agents. Observe that, because Zi ≥ 0

for all i, we have αi ≥ 0 for all i.

Suppose that the public randomization device generates a uniform distribution over

J ⊆ NQ : |J | = sQ. Given that the realization of the public randomization device is J ,

for each i ∈ P and k ∈ J , let xik = αiYk. That is, we let each proposer i make an offer to a

non-proposer k in the subset of the non-proposers J that gives to the non-proposer k in J a

share αi of the sum of his previous offers Yk.

Note that for each i ∈ P , the net transfers with the original offers are −Zi = Yi −Xi.

Observe also that we have∑

j∈P Xj =∑

i Yi, that is, the sum of all offers made by the

proposers to the other agents must be equal to the sum of the offers received by all agents.

This implies that∑

j∈P Zj =∑

k∈J Yk, that is, the sum of the net offers made by the

proposers to the other agents is equal to the sum of the offers received by the non-proposers

in the subset J .

By construction, the sum of the offers received by each agent i ∈ J is the same under the

39

previous offers and under the new offers. Under the new offers, the net offers made by each

agent i ∈ P to the other agents are −∑

k∈J xik = −αi∑

k∈J Yk = − Zi∑j∈P Zj

∑k∈J Yk = −Zi

because∑

j∈P Zj =∑

k∈J Yk. Therefore, the new set of offers gives the same payoffs to all

agents. Thus it is consistent with equilibrium for the agents to make and accept these offers

under Q.

Lemma 2. Fix a distribution of proposal power f . Consider the class of pure strategy

symmetric stationary subgame perfect equilibria in which agents use stage-undominated

voting strategies and proposers do not make net offers that leave them strictly less than

their discounted equilibrium continuation values. If P = Q(σ′) for some equilibrium σ′ that

exists given f such that some proposer makes strictly positive offers under Q 6∈ Q(σ′), then

(given f) there exists an equilibrium σ such that P = Q(σ) and all proposers make zero

offers under Q 6∈ Q(σ).

Proof of lemma 2.

Fix Q 6∈ Q(σ′) such that some proposer makes strictly positive offers when the

proposing partition is Q. Let x denote the sum of offers that an agent in the minimal

non-proposing winning coalition (MNWC) receives from the proposers under Q.

Note that the requirement that proposers do not make net offers that leave them strictly

less than their discounted equilibrium continuation values, combined with the requirement

that agents use stage-undominated voting strategies, implies that the proposers always vote

in favor of the proposals. Then, because there is disagreement under Q in σ′ and agents use

stage-undominated voting strategies, we must have x < δV σ′for an agent in the MNWC

(note that this inequality must be satisfied for all members of the MNWC because all

members of this coalition are treated symmetrically by proposers).

Let us use y to denote the offer of a proposer on DQ pies to a member of MNWC in

equilibrium σ′. Then, in order to achieve agreement, a proposer on DQ pies would have to

give δV σ′−(x−y) (rather than y) to each of the sQ of the members of the MNWC. By lemma

1, for the purposes of determining the payoffs that are attainable in equilibrium, it is without

loss of generality to focus on the equilibria in which no proposer receives strictly positive

offers. Then, because no proposer receives strictly positive offers, a proposer on DQ pies keeps

DQ less the offers she makes to the non-proposers. Then disagreement under Q in σ′ implies

that we must have DQ−sQ(δV σ′−(x−y)) < δV σ′, that is, a proposer on DQ pies is unwilling

to make the offers leading to an agreement. Because DQ−sQδV σ′ ≤ DQ−sQ(δV σ′−(x−y)),

the fact that DQ − sQ(δV σ′ − (x− y)) < δV σ′implies that DQ − sQδV σ′

< δV σ′.

By lemma 7, in any equilibria σ and σ′ satisfyingQ(σ) = Q(σ′) we must have V σ′= V σ.

Then, because V σ′= V σ, the fact that DQ−sQδV σ′

< δV σ′implies that DQ−sQδV σ < δV σ.

Therefore, there must also be disagreement under Q in equilibrium σ (given the conjecture

40

that all other proposers make zero offers).

Let the strategies in σ under proposing partitions Q ∈ Q(σ) be the same as the

strategies in σ′ under proposing partitions Q ∈ Q(σ′). Observe that, because V σ′= V σ,

the incentives to agree under Q ∈ Q(σ) are preserved, so for all Q ∈ Q(σ) we must have

Q ∈ Q(σ′).

Lemma 3. Consider a pure strategy equilibrium σ satisfying the following:

1. the equilibrium is symmetric, stationary and subgame perfect

2. agents use stage-undominated voting strategies

3. all proposers make zero offers when the proposing partition is Q 6∈ Q(σ)

Then, generically, this equilibrium is proposal stage undominated if and only if for all Q 6∈Q(σ), we have δV σ ≤ m−DQ

q−1.

Proof of lemma 3.

We first argue that if for all Q 6∈ Q(σ), we have δV σ ≤ m−DQn−1

, then the equilibrium is

proposal stage undominated. Because of the requirement that agents use stage-undominated

voting strategies, it is sufficient to argue that the strategies used by the proposers (in proposal

stage games) are weakly undominated.

Consider an equilibrium σ and a proposing partition Q such that Q 6∈ Q(σ). Recall that

an equilibrium σ requires that all proposers make zero offers when the proposing partition

is Q 6∈ Q(σ). Fix a stage game and consider any strategy σ′i of proposer i that involves

making strictly positive offers summing to x > 0 in this stage game under Q. Let ki denote

the number of pies that proposer i controls.

Let NQ denote the set of the non-proposers under the proposing partition Q. Suppose

that the public randomization device generates a uniform distribution over J ⊆ NQ : |J | =sQ. Suppose also that proposers other than proposer i use a profile of strategies σ′′−i that

has the following feature: if the realization of the public randomization device is J , each

proposer j offers to each non-proposer in J the amountkj−δV σsQ

, where kj denotes the number

of pies that the proposer controls. Then each non-proposer in J receives offers summing tom−ki−(|Q|−1)δV σ

sQ. Because each agent is using stage-undominated voting strategies, if

m− ki − (|Q| − 1) δV σ

sQ≥ δV σ (1)

then the offers will be accepted by the non-proposers in the response stage game that follows.

The offers will also be accepted by the proposers because in the event of agreement each

proposer receives δV σ.

41

Observe that if sQ = q − |Q|, then (1) is equivalent to δV σ ≤ m−kiq−1

. If, on the other

hand, sQ = 0, then Q ∈ Q(σ). Therefore, for all Q 6∈ Q(σ), (1) is equivalent to δV σ ≤ m−kiq−1

.

Because DQ ≥ ki for all ki ∈ Q, in order to ensure that (1) holds for all ki ∈ Q, it is sufficient

to have δV σ ≤ m−DQq−1

. The last condition in the lemma ensures that δV σ ≤ m−DQq−1

holds.

Thus the payoff of proposer i (in the proposal stage game under consideration) from

using the strategy σ′i is ki − x, while the payoff to using her equilibrium strategy (provided

that the other agents are using the strategies σ′′−i) is ki. Therefore, any strategy of a proposer i

that involves making strictly positive offers under Q cannot weakly dominate her equilibrium

strategy (that involves making zero offers under Q).

Next, consider a proposing partition Q such that Q ∈ Q(σ). Suppose that the

equilibrium prescribes that proposer i makes offers summing to x under Q. Fix a stage

game. Consider any strategy σ′i of proposer i that involves making offers that sum to strictly

less than x. Let U denote the payoff of the proposer from the agreement in this stage game

under Q. Because there is agreement under Q in the equilibrium σ, we must have U ≥ δV σ.

Moreover, generically, U ≥ δV σ implies that U > δV σ.

Suppose that the other proposers use their equilibrium strategies σ−i when proposer i

uses the strategy σ′i. Observe that this implies that there exists at least one non-proposer

who strictly prefers to reject the offers, so that the strategy σ′i leads to disagreement in this

stage game under Q. Because U > δV σ, the strategy σ′i yields a strictly lower payoff to

proposer i than strategy σi provided that the other proposers use strategies σ−i. Therefore,

σ′i cannot weakly dominate σi.

Finally, consider any strategy σ′i of proposer i that involves making offers that sum to

strictly more than x. Suppose that the other proposers use their equilibrium strategies σ−iwhen proposer i uses the strategy σ′i. Then both the strategy σi and the strategy σ′i lead to

agreement in this stage game under Q but σ′i yields a strictly lower payoff to the proposer

than σi (because the sum of the offers that proposer i makes is strictly greater under σ′i).

Therefore, σ′i cannot weakly dominate σi.

We next argue that if for some Q 6∈ Q(σ) we have δV σ >m−DQq−1

, then the equilibrium

is proposal stage dominated. Consider an equilibrium σ and a proposing partition Q such

that Q 6∈ Q(σ) and δV σ >m−DQq−1

. Let ε = 12

(m− nδV σ). Let i denote a proposer that

controls DQ pies. Fix a stage game and consider the strategy σ′i of proposer i that has the

following feature: if the realization of the public randomization device is J , then i offers to

each non-proposer in J the amountDQ−δV σ−ε

sQ. Suppose that the other proposers use a profile

of strategies σ′′−i that has the following feature: each proposer j offers to each non-proposer

in J the amountkj−δV σsQ

. Then each non-proposer in J receives offers summing to m−|Q|δV σ−εsQ

.

42

Because each agent is using stage-undominated voting strategies, if

m− |Q|δV σ − εsQ

≥ δV σ (2)

then the offers will be accepted by the non-proposers in the response stage game that follows.

Observe that if sQ = q − |Q|, then (2) is equivalent to ε ≤ m− qδV σ, while if sQ = 0,

then Q ∈ Q(σ). Therefore, for all Q 6∈ Q(σ), (2) is equivalent to ε ≤ m − qδV σ. Then the

fact that δV σ < mn≤ m

qand the definition of ε imply that (2) is satisfied. The offers will

also be accepted by the proposers because in the event of agreement each proposer receives

at least δV σ.

On the other hand, if proposer i uses her equilibrium strategy σi that prescribes making

zero offers under Q, then, because δV σ >m−DQq−1

, the argument above implies that there

does not exist a strategy profile σ′′−i for the other proposers that ensures acceptance in the

response stage. Thus the equilibrium strategy yields δV σ, while σ′i yields no less than δV σ for

all feasible strategies of the other proposers in the proposal stage game and yields δV σ + ε to

proposer i given that the other proposers use strategies σ′′−i. This implies that σi is proposal

stage dominated by σ′i.

Lemma 4. If there exists a distribution f such that given f there is a proposal

stage undominated equilibrium with agreement set Q(σ) ⊂ Q, then maxQ6∈Q(σ)DQsQ+1

<

minQ 6∈Q(σ)m−DQq−1

.

Proof of lemma 4.

Part 1 of Theorem 2 in section 5.4 implies that if there exists an equilibrium σ with

an agreement set Q(σ) ∈ Ω(f), thenDQsQ+1

< δV σ for all Q 6∈ Q(σ). Combining this

with the result in lemma 3 that δV σ ≤ m−DQq−1

for all Q 6∈ Q(σ) is necessary for an

equilibrium σ to be proposal stage undominated, we find that if there exists a proposal

stage undominated equilibrium σ, then δV σ ∈(

DQsQ+1

,m−DQq−1

)for all Q 6∈ Q(σ). This implies

that ∩Q6∈Q(σ)

(DQsQ+1

,m−DQq−1

)6= ∅. Therefore, maxQ6∈Q(σ)

DQsQ+1

< minQ6∈Q(σ)m−DQq−1

must be

satisfied, as required.

Lemma 5. Suppose that |k ∈ Q : k = DQ| = 1 and q ≥ m. Then the order on the space

of the proposing partitions has the following property: if

Q \DQ = Q′ \ k1, k2

for some k1, k2 ∈ Q′ such that k1 + k2 = DQ, thenDQsQ+1

≥ DQ′

sQ′+1.

Proof of lemma 5.

43

By construction, we have DQ > k for all k ∈ Q′. This implies that DQ′ = DQ − t for

some t ≥ 1. Moreover, by construction, we have |Q′| = |Q| + 1, so that sQ′ = sQ − z for

some z ∈ 0, 1. ThenDQ′

sQ′+1=

DQ−tsQ−z+1

for some t ≥ 1, z ∈ 0, 1.

It is sufficient to show thatDQ−tsQ−z+1

≤ DQsQ+1

. This is equivalent to DQz ≤ t(sQ + 1).

Because t ≥ 1 and sQ + 1 > 0, it is sufficient to show that DQz ≤ sQ + 1. If z = 0, we are

done. Then, since z ∈ 0, 1, it is sufficient to show that DQ ≤ sQ + 1. Because q ≥ m

implies that sQ = q − |Q|, this is equivalent to showing that DQ ≤ q − |Q|+ 1.

Observe that, because |Q| ≤ m−DQ, we have q − |Q|+ 1 ≥ q + 1 +DQ −m. Then it

is sufficient to show that DQ ≤ q + 1 +DQ −m. This is equivalent to q−m ≥ −1. Because

q ≥ m, this is satisfied.

Lemma 6 (Necessary Conditions for an Equilibrium). Suppose that (Q1,Q2) is a

partition of Q and there exists an equilibrium σ such that Q(σ) = Q2. Then δV σ >DQsQ+1

for

all Q ∈ Q1.

Moreover, if σ is a simple equilibrium and for all Q ∈ Q we have that f(Q) > 0 implies

that |k ∈ Q : k = DQ| = 1, then δV σ <DQsQ+1

for all Q ∈ Q2.

Proof of lemma 6.

Because Q(σ) = Q2, it must be the case that if Q ∈ Q1, then each proposer on k ∈ Qpies prefers to make zero offers given that all other proposers in the proposing partition Q are

making zero offers. The payoff to making the offers that are acceptable to the non-proposers

is k − sQδV σ, while the payoff to making zero offers is δV σ. Thus it must be the case that

k − sQδV σ < δV σ, which is equivalent to δV σ > rsQ+1

for all k ∈ Q. Because DQ ∈ Q, it

must be the case that, in particular, δV σ >DQsQ+1

.

Suppose that σ is a simple equilibrium and for all Q ∈ Q we have that f(Q) > 0

implies that |q ∈ Q : q = DQ| = 1. This implies that for each proposing partition Q ∈ Q2

there is a unique agent who is a proposer on DQ pies. Because Q \ Q(σ) = Q1 and σ is a

simple equilibrium, it must be the case that if Q ∈ Q2, then the proposer on DQ pies prefers

to make an offer of δV σ to each of the sQ agents in a subset of non-proposers given that all

other proposers in the proposing partition Q are making zero offers. The payoff to making

these offers is DQ−sQδV σ, while the payoff to offering anything strictly less than δV σ to any

non-proposer is δV σ. Thus it must be the case that DQ− sQδV σ > δV σ, which is equivalent

to δV σ <DQsQ+1

, as required.

Lemma 7. Suppose that the distribution of proposal power is F and there exists an

44

equilibrium σ. The value in an equilibrium σ with Q(σ) = P is given by

U(P , f) =m

n

∑Q∈P f(Q)

1− δ + δ∑

Q∈P f(Q)

Proof of lemma 7.

Consider an equilibrium in which there is agreement if and only if Q ∈ P . Because

under every proposing partition every agent has the same probability of being a proposer,

if the agents use symmetric strategies in equilibrium, then their equilibrium values are the

same.

Thus, because in a symmetric equilibrium the expected payoff in every state in which

there is agreement must be the same for every agent and must equal the available surplus

(which is m) divided by the number of agents (which is n), the value function V must satisfy

V =m

n

∑Q∈P

f(Q) + δ

(1−

∑Q∈P

f(Q)

)V

This is equivalent to

V =m

n

∑Q∈P f(Q)

1− δ + δ∑

Q∈P f(Q)

as required.

Lemma 8 (Sufficient Conditions for an Equilibrium). Suppose that (Q1,Q2) is a

partition of Q, there exists a constant κ > 0 satisfying δκ >DQsQ+1

for Q ∈ Q1 and δκ <DQsQ+1

for Q ∈ Q2, and there exists a proposal power distribution f such that κ = U(Q2, f).

Then there exists a simple equilibrium σ such that Q(σ) = Q2. Moreover, the value in the

equilibrium σ is given by U(Q2, f).

Proof of lemma 8.

We will show that there exists a simple equilibrium σ such that Q(σ) = Q2. By

lemma 7, the value in an equilibrium σ is given by U(Q(σ), f). Because, by the hypothesis,

there exists a proposal power distribution f such that κ = U(Q2, f), κ is the value in the

equilibrium σ.

We first show that if Q ∈ Q1, then each proposer on k ∈ Q pies prefers to make zero

offers given that all other proposers in the proposing partition Q are making zero offers.

The payoff to making the offers that are acceptable to sQ of the non-proposers is k − sQδκ,

while the payoff to making zero offers is δκ. Thus we require that k − sQδκ < δκ, which is

equivalent to δκ > ksQ+1

for all k ∈ Q. Because DQ ≥ k for all k ∈ Q, the fact that δκ >DQsQ+1

45

for all Q ∈ Q1 implies that δκ > ksQ+1

for all k ∈ Q, Q ∈ Q1 is satisfied, as required.

Let NQ denote the set of the non-proposers under Q. After the proposing partition

Q is realized, the public randomization device generates a distribution that is uniform on

J ⊆ NQ : |J | = sQ. Let K = |k ∈ Q : K = DQ|. If the realization of the public

randomization device is J ⊆ NQ, then each proposer on DQ pies makes an offer of 1Kδκ to

each agent who is in J and makes zero offers to all other agents.

We next show that if Q ∈ Q2, then each of the K agents who is a proposer on DQ pies

weakly prefers to make an offer of 1Kδκ to each agent who is in J . The sum of offers each

proposer on DQ pies makes issQKδκ. The payoff to making the required offers is DQ− sQ

Kδκ,

while the payoff to deviating to offering a strictly smaller amount to any agent in J is δκ.

Thus we require that DQ− sQKδκ > δκ, which is equivalent to δκ <

DQsQK

+1. Because δκ <

DQsQ+1

for all Q ∈ Q2, we have δκ <DQsQ+1

≤ DQsQK

+1, as required.

Observe that the equilibrium σ is simple because only the proposers on the maximum

number of pies DQ make strictly positive offers.

Lemma 9 (Worst Equilibrium). Let σ denote an equilibrium such that Q(σ) = Cf and

Q(σ) ∈ Ω(f).30 If for some equilibrium σ′ we have that Q(σ′) ∈ Ω(f) and Q(σ′) 6= Q(σ),

then V σ′> V σ.

Proof of lemma 9.

Suppose that there exists an equilibrium σ′ such that Q(σ′) ∈ Ω(f) and Q(σ′) 6= Q(σ).

If Q(σ) ⊂ Q(σ′), then we are done. Therefore, suppose that there exists Q′ ∈ Q(σ) such that

Q′ 6∈ Q(σ′). By lemma 6, a necessary condition to have Q(σ′) ∈ Ω(f) is that δV σ′>

DQsQ+1

for all Q 6∈ Q(σ′). In particular, because Q′ 6∈ Q(σ′), we must have

δV σ′>

DQ′

sQ′ + 1(3)

Recall that Qf = min Q : Q ∈ Cf, where the minimum is taken with respect to the

order induced by RQ. Recall also that BQ = 1 − 1−δ

δ

(mn

sQ+1

DQ−1

) for Q such thatDQsQ+1

< mn

,

BQ = −∞ for Q such thatDQsQ+1

≥ mn

, U (Cf , f) = mn

1−F (Qf )

1−δF (Qf ), V σ = U(Cf , f) and Q ∈ Cf if

and only if F (Q) ≥ BQ. Then F (Qf ) ≥ BQf implies that

DQf

sQf + 1≥ δV σ (4)

30The fact that such an equilibrium exists follows from the proof of part 1 of Proposition 4.

46

Note that, because Q′ ∈ Q(σ), we have RQ′ ≥ RQf , that is,

DQ′

sQ′ + 1≥

DQf

sQf + 1(5)

Thus δV σ′>

DQ′

sQ′+1≥

DQfsQf+1

≥ δV σ, where the first inequality follows from (3), the

second inequality follows from (5) and the third inequality follows from (4). Therefore,

δV σ′> δV σ, as required.

Proposition 4.

1. there exists an equilibrium

2. σ is the worst equilibrium if and only if the set of the proposing partitions under which

there is agreement in the equilibrium σ is given by

Q(σ) = Cf

3. there exists a worst equilibrium that is simple

4. if for all Q, f(Q) > 0 implies that |k ∈ Q : k = DQ| = 1, then there exists a unique

simple equilibrium

Proof of Proposition 4.

We will prove the first part of the Proposition first. We will show that there exists a

simple equilibrium σ such that Q(σ) = Cf and Q(σ) ∈ Ω(f). Then the second part of the

Proposition (the claim that this is the worst equilibrium) will follow from lemma 9 and the

third part of the Proposition (the claim that there exists a worst equilibrium that is simple)

will follow from the first and the second parts of the Proposition.

Lemma 8 implies that if δU(Q(σ), f) >DQsQ+1

for all Q 6∈ Q(σ) and δU(Q(σ), f) <DQsQ+1

for all Q ∈ Q(σ), then there exists a simple equilibrium σ such that Q(σ) ∈ Ω(f). By

definition of Cf , we have δU(Cf , f) >DQsQ+1

for all Q 6∈ Cf and δU(Cf , f) <DQsQ+1

for all Q ∈ Cf .Therefore, there exists a simple equilibrium σ such that Q(σ) = Cf and Q(σ) ∈ Ω(f), as

required.

Lemma 6 implies that if for all Q we have that f(Q) > 0 implies that |k ∈ Q : k =

DQ| = 1, then if σ is a simple equilibrium, it must be the case that δU(Q(σ), f) >DQsQ+1

for all Q 6∈ Q(σ) and δU(Q(σ), f) <DQsQ+1

for all Q ∈ Q(σ). By definition of Cf , this is

equivalent to Q(σ) = Cf . Therefore, the agreement set Q(σ) for the simple equilibrium is

unique, which implies that the simple equilibrium is unique.

Proof of Theorem 1.

47

Follows from the second part of Proposition 4.

Proof of Proposition 1.

Follows from the third and fourth parts of Proposition 4.

Proof of Proposition 2.

We make the dependence of the number of supporters sQ on q explicit by writing sqQ.

Recall that, by assumption, q > m for all q, which implies that sqQ = q−|Q| > 0 for all Q and

q. Define Jq,pQ = δmn

p1−δ+δp −

DQsqQ+1

. Observe that if for all Q and q1, q2 we have sq1Q > 0 and

sq2Q > 0, then whenever q1 > q2, we have sq1Q > sq2Q . This implies that ∂Jq,pQ /∂q > 0. Observe

also that p 7→ δmn

p1−δ+δp is strictly increasing. This implies that ∂Jq,pQ /∂p < 0. Letting p(q)

be implicitly defined by Jq,p(q)Q = 0, observe that then we have ∂p(q)

∂q= −∂Jq,pQ /∂q

∂Jq,pQ /∂p< 0.

By Proposition 4, the agreement set in the worst equilibrium is given by Cqf =Q : δm

n

∑Q∈Q(σq)

f(Q)

1−δ+δ∑Q∈Q(σq)

f(Q)− DQ

sqQ+1≤ 0

, where I have made the dependence of the worst

equilibrium agreement set on q explicit. Therefore, Cqf =Q : Jq,pQ ≤ 0

for p =∑

Q∈Q(σq)f(Q). Then the fact that ∂p(q)

∂q< 0 implies that

∑Q∈Q(σq1 ) f(Q) ≤

∑Q∈Q(σq2 ) f(Q),

as required.

Definition 2. Given P ⊂ Q, we say that L(P) is the disagreement partition if

L(P) = maxQ : Q ∈ Q \ P

where the maximum is taken with respect to the order on the proposing partitions induced

by RQ.

Definition 2 says that, given an equilibrium agreement set P ⊂ Q,31 L(P) is the

disagreement partition if it is the highest partition according to the order induced by RQ

that is not in the agreement set.

Lemma 10. If∑

Q∈P f(Q) > 1−BL(P) or P = Q, then P ∈ Ω(f).

Proof of lemma 10.

We will show that there exists an equilibrium σ such that Q(σ) = P . Suppose first

that P 6= Q. We require that there is disagreement under the proposing partitions not in

P . We conjecture equilibrium strategies such that under the proposing partitions not in P ,

all proposers make zero offers. For this, it is sufficient that, for each Q ∈ Q \ P , provided

that all other proposers are making zero offers, a proposer on DQ pies prefers not to make

the offers leading to an agreement. In order to obtain agreement, a proposer on DQ pies

would need to make offers yielding a payoff no greater than DQ − sQδV σ to the proposer,

31Note that, because the fact that P ⊂ Q implies that Q \ P 6= ∅, L(P) is well-defined.

48

while the payoff to disagreeing is δV σ. Thus it is sufficient to have DQ − sQδV σ < δV σ for

all Q ∈ Q \ P , which is equivalent to δV σ >DQsQ+1

for all Q ∈ Q \ P .

Recall that L(P) = maxQ : Q ∈ Q \ P, where the maximum is taken with respect

to the order induced by RQ. Because, by lemma 7, we have V σ = U(P , f),∑

Q∈P f(Q) >

1 − BL(P) is equivalent to δV σ >DL(P)

sL(P)+1. Because RL(P) ≥ RQ for all Q 6∈ P , this implies

that δV σ >DQsQ+1

for all Q 6∈ P .

Suppose next that P = Q. Then Q \ P = ∅, and it is sufficient to show that there is

agreement under the proposing partitions in P .

Given Q ∈ Q and k ∈ Q, define

βk(Q) =kn−msQm

We require that there is agreement under the proposing partitions in P . For each

Q ∈ P , we conjecture the following strategies. The public randomization device generates

a distribution that is uniform on J ⊆ NQ : |J | = sQ, where NQ denotes the set of the

non-proposers under the proposing partition Q. If the realization of the public randomization

device is J ⊆ NQ, then each proposer controlling k pies in Q offers βk(Q)δV σ to each agent

in J . Observe that βk(Q) > 0 and

∑k∈Q

βk(Q) =

∑k∈Q kn

sQm− |Q|sQ

=mn

sQm− n− sQ

sQ= 1

Then each non-proposer receives the sum of offers equal to∑

k∈Q βk(Q)δV σ = δV σ and

accepts these offers.

The payoff to making these offers is k − sQβk(Q)δV σ, while the payoff to deviating to

offers that are any lower is δV σ. Thus we require that k − sQβk(Q)δV σ > δV σ, which is

equivalent to δV σ < ksQβk(Q)+1

. Because δV σ < mn

, it is sufficient to show that

m

n≤ k

sQβk(Q) + 1

The definition of βk(Q) ensures that this constraint holds with equality.

Lemma 11. If P ∈ Ω(f), then for all P ⊆ Q such that P ⊂ P, we have P ∈ Ω(f).

Proof of lemma 11.

Suppose first that P = Q. Then the result follows from lemma 10.

49

Suppose next that P ⊂ Q. Then Q \ P 6= ∅, so L(P) is well-defined.

Because P ∈ Ω(f), there exists an equilibrium σ such that Q(σ) = P . Then there is

disagreement in equilibrium σ under proposing partitions in Q\P . Because P ⊂ P , we have

Q \ P ⊂ Q \ P . This implies that there is disagreement in equilibrium σ under proposing

partitions in Q \ P .

Because there is disagreement in equilibrium σ under proposing partitions in Q \ P ,

lemma 6 implies that δV σ > RQ for all Q ∈ Q \ P . Because V σ = U(Q(σ), f), this is

equivalent to ∑Q∈P

f(Q) > 1−BL(P) (6)

Because P ⊂ P , we have that L(P) ≤ L(P)

. Because Q 7→ BQ is decreasing, this

implies that

1−BL(P) ≤ 1−BL(P) (7)

Because P ⊂ P , we have ∑Q∈P

f(Q) ≥∑Q∈P

f(Q) (8)

Then ∑Q∈P

f(Q) ≥∑Q∈P

f(Q) > 1−BL(P) ≥ 1−BL(P)

where the first inequality follows from (8), the second inequality follows from (6) and the

third inequality follows from (7).

Thus∑

Q∈P f(Q) > 1−BL(P) holds. Then the result follows from lemma 10.

Proof of Theorem 2.

Proof of Part 1 of the Theorem.

The claim that Q ∈ Ω(f) follows from lemma 10.

Next consider P ⊂ Q. We will show that if P ⊂ Q, then P ∈ Ω(f) if and only if∑Q∈P f(Q) > 1−BL(P). Part 1 of the Theorem will then follow from this.

Suppose that P ∈ Ω(f). By lemma 6, a necessary condition to have P ∈ Ω(f) is

that δU(P , f) > RQ for all Q 6∈ P . In particular, we must have δU(P , f) >DL(P)

sL(P)+1. This

implies that∑

Q∈P f(Q) > 1−BL(P), as required. Conversely, suppose that∑

Q∈Q(σ) f(Q) >

1−BL(Q(σ)). Then the result follows from lemma 10.

Proof of Part 2 of the Theorem.

50

Lemma 6 implies that if δV (RQ) ≤ RQ, then agents must agree at Q. Note that the

fact that V (RQ) ≤ mn

for all Q implies that δV (RQ) ≤ δmn

. Then for Q such that δmn≤ RQ,

we have δV (RQ) ≤ RQ. This implies that at all Q such that δmn≤ RQ, agents must agree in

any equilibrium.

Proof of Part 3 of the Theorem.

Proposition 4 implies that Cf ∈ Ω(f). The result then follows from lemma 11.

Proof of Part 4 of the Theorem.

Suppose that∑

Q 6∈Cf f(Q) < f(Q′) for all Q′ ∈ Cf \ G. Part 3 of the Theorem shows

that Q ⊆ Q : Cf ⊆ P ⊆ Ω(f). Given this, it is sufficient to show that Ω(f) ⊆ Q ⊆ Q :

Cf ⊆ P.

Then we would like to show that there does not exist P ∈ Ω(f) such that Q′ 6∈ P for

some Q′ ∈ Cf . Thus we fix P such that there exists Q′ satisfying Q′ 6∈ P and Q′ ∈ Cf . We

will show that P 6∈ Ω(f).

Because Q′ 6∈ P , we have L(P) ≥ Q′. Because Q 7→ BQ is decreasing, this implies that

1−BL(P) ≥ 1−BQ′ (9)

Observe that∑

Q∈P f(Q) ≤∑

Q 6=Q′ f(Q) = 1 − f(Q′). Because∑

Q6∈Cf f(Q) < f(Q′)

by the hypothesis, we have 1− f(Q′) < 1−∑

Q 6∈Cf f(Q) =∑

Q∈Cf f(Q). This implies that∑Q∈P

f(Q) <∑Q∈Cf

f(Q) (10)

Recall that the definition of Cf implies that δU(Cf , f) < minQ∈Cf RQ and that Qf =

minQ : Q ∈ Cf, where the minimum is taken with respect to the order induced by RQ.

Then δU(Cf , f) < minQ∈Cf RQ implies that∑Q∈Cf

f(Q) < 1−BQf (11)

Therefore,∑

Q∈P f(Q) <∑

Q∈Cf f(Q) < 1 − BQf ≤ 1 − BQ′ ≤ 1 − BL(P), where

the first inequality follows from (10), the second inequality follows from (11), the third

inequality follows from the fact that Q′ ≥ Qf since Q′ ∈ Cf and from the fact that Q 7→ BQ

is decreasing, and the fourth inequality follows from (9).

Thus we have∑

Q∈P f(Q) < 1 − BL(P). Then part 1 of the Theorem implies that

P 6∈ Ω(f), as required.

51

Given any proposing partition Q such that Q′ : RQ′ < RQ 6= ∅, define

Q− = maxQ′ : RQ′ < RQ

where the maximum is defined with respect to the order induced by RQ. Q− is the proposing

partition that precedes Q according to the order induced by RQ.

Lemma 12. If P ∈ Ω(f), P ∈ Ω(f)

, P 6= Q, P 6= Q and P = Q′ : RQ′ ≥ RQ,P =

Q′ : RQ′ ≥ RQ

, then

F (Q−) < F(Q−)

if and only if U(P , f) > U(P , F

)Proof of lemma 12.

Note that, because P 6= Q and P 6= Q, Q− and Q− are well-defined. Observe that,

because P = Q′ : RQ′ ≥ RQ and P =Q′ : RQ′ ≥ RQ

, we have

∑Q′∈P f(Q′) = 1 −

F (Q−) and∑

Q′∈P f(Q′) = 1 − F(Q−)

. Lemma 7 implies that U(P , f) > U(P , F

)if

and only if∑

Q′∈P f(Q′) >∑

Q′∈P f(Q′). Thus F(Q−)< F (Q−) if and only if U(P , f) >

U(P , F

).

Proof of Proposition 3.

We will prove the following version of the Proposition (note that here we are asserting

that weak inequalities can be replaced with strict inequalities under certain conditions):

1. Suppose that F dominates F in the sense of FOSD. Then

(a) Cf ⊆ Cf

(b) if Cf 6= Q and minQ∈Q\m

(F (Q)− F (Q)

)> 0, then there exists ε > 0 such that

if maxQ

(F (Q)− F (Q)

)< ε , then

minP∈Ω(f)

U(P , f

)< minP∈Ω(f)

U(P , f)

2. If Cf 6= Q, then for all δ ∈ (0, 1) and m, n, there exist δ and M > 0 such that if

δ ∈(δ − δ, δ

)and m

n∈(mn−M, m

n

), the payoff in the worst equilibrium

(a) is strictly higher under δ than under δ and

(b) is strictly higher under m and n than under m and n

Because F dominates F in the sense of FOSD, we have F (Q) ≥ F (Q) for all Q. Let V

52

denote the value function (in the worst equilibrium) provided that the distribution of power

is f . Note that F (Q) ≥ F (Q) for all Q implies that V (RQ) ≤ V (RQ) for all Q. Then,

because Cf = Q : RQ ≥ δV (RQ), we have Cf ⊆ Cf , as required.

We prove the second part of the Proposition in two steps. First, we show that the

distance maxQ

(F (Q)− F (Q)

)between F and F can be chosen to be sufficiently small

such that the worst equilibrium agreement set under f is the same as the worst equilibrium

agreement set under f . Then we show that the payoff in the worst equilibrium under f is

strictly lower than under f .

Because Cf 6= Q and Cf ⊆ Cf , we have Cf 6= Q. Then Q \ Cf 6= ∅, which implies that

L(Cf ) is well-defined.

We choose

ε = BL(Cf ) − F (L(Cf ))

Note that the definitions of Cf , L(Cf ) and BQ imply that ε > 0.

Observe that if maxQ

(F (Q)− F (Q)

)< ε , then the definition of ε implies that F (Q)−

F (Q) < BL(Cf )−F (L(Cf )) for all Q. In particular, we have F (L(Cf ))−F (L(Cf )) < BL(Cf )−F (L(Cf )), which implies that

F (L(Cf )) < BL(Cf ) (12)

For all Q ≤ L(Cf ) we have F (Q) ≤ F (L(Cf )) < BL(Cf ) ≤ BQ, where the first inequality

follows from the fact that F is increasing, the second inequality follows from (12) and the

third inequality follows from the fact that Q 7→ BQ is decreasing.

Therefore, for all Q ≤ L(Cf ) we have F (Q) < BQ. Then the definition of Cf implies

that Q 6∈ Cf for all Q ≤ L(Cf ). Equivalently, Cf ⊆ Cf .

Observe that the fact that Cf ⊆ Cf and part 1 of the Proposition imply that

Cf = Cf (13)

Because Cf = Q′ : RQ′ ≥ RQ for some Q ∈ Q and Q \ Cf 6= ∅, lemma 12 applies.

Lemma 12 implies that F (L(Cf )) < F (L(Cf )) if and only if

U(Cf , f) > U(Cf , f

)(14)

Note that, because L(Cf ) 6∈ Cf and Theorem 2 implies that m ∈ G and Q ⊆ Cf ′ for

all PMFs f ′, it must be the case that L(Cf ) 6= m.

53

Because L(Cf ) 6= m, minQ∈Q\m

(F (Q)− F (Q)

)> 0 implies that F (L(Cf )) <

F (L(Cf )) holds. Then (13) and (14) imply that U(Cf , f) > U(Cf , f

).

Observe that lemma 9 implies that Cf = arg minP∈Ω(f) U(P , f) and Cf =

arg minP∈Ω(f) U(P , f

). Then the fact that U(Cf , f) > U

(Cf , f

)implies that

minP∈Ω(f) U(P , f

)> minP∈Ω(f) U(P , f), as required.

Making the dependence of BQ and Cf on δ and mn

explicit, let us write Bδ,mn

Q and

Cδ,mn

f . Let ε = Bδ,mn

L

(Cδ,mnf

) − F(L(Cδ,

mn

f

)). Observe that there exists δ > 0 such that if

δ ∈(δ − δ, δ

), then B

δ,mn

L

(Cδ,mnf

) − Bδ,mn

L

(Cδ,mnf

) ∈ (0, ε) and there exists M > 0 such that if

mn∈(mn−M, m

n

), then B

δ,mn

L

(Cδ,mnf

) − Bδ,mn

L

(Cδ,mnf

) ∈ (0, ε). Then Bδ,mn

L

(Cδ,mnf

) > F(L(Cδ,

mn

f

))and B

δ,mn

L

(Cδ,mnf

) > F(L(Cδ,

mn

f

)). Moreover, B

δ,mn

Q < Bδ,mn

Q ≤ F (Q) and Bδ,mn

Q < Bδ,mn

Q ≤

F (Q) for all Q > L(Cδ,

mn

f

), where the second inequality follows from the definition of

L(Cδ,

mn

f

). This implies that Cδ,

mn

f = Cδ,mn

f and Cδ,mn

f = Cδ,mn

f . Then U(Cδ,

mn

f , f)

=

mn

∑Q∈C

δ,mnf

f(Q)

1−δ+δ∑Q∈C

δ,mnf

f(Q)> m

n

∑Q∈C

δ,mnf

f(Q)

1−δ+δ∑Q∈C

δ,mnf

f(Q)= m

n

∑Q∈C

δ,mnf

f(Q)

1−δ+δ∑Q∈C

δ,mnf

f(Q)= U

(Cδ,

mn

f , f)

and

U(Cδ,

mn

f , f)

= mn

∑Q∈C

δ,mnf

f(Q)

1−δ+δ∑Q∈C

δ,mnf

f(Q)> m

n

∑Q∈C

δ,mnf

f(Q)

1−δ+δ∑Q∈C

δ,mnf

f(Q)= m

n

∑Q∈C

δ,mnF

f(Q)

1−δ+δ∑Q∈C

δ,mnf

f(Q)=

U(Cδ,

mn

f , f)

. Thus U(Cδ,

mn

f , f)

> U(Cδ,

mn

f , f)

and U(Cδ,

mn

f , f)

> U(Cδ,

mn

f , f)

, as

required.

Proof of Theorem 3.

Claim 4.1. Fix Q ∈ Q such that Cf = Q′ : RQ′ ≥RQ for some F ∈ G(F , F ) and Q\Cf 6= ∅.

Then

F (L(Cf )) ≥ ΦQ,F,F (L(CφQ,F,F ))

Proof of claim 4.1.

Observe that, because Q \ Cf 6= ∅, L(Cf ) is well-defined. Because CφQ,F,F = Cf ,L(CφQ,F,F ) is well-defined.

We have F (L(Cf )) = F(L(CφQ,F,F

))≥ F

(L(CφQ,F,F

))= ΦQ,F,F

(L(CφQ,F,F

)), where

the first equality follows from the fact that CφQ,F,F = Cf = Q′ : RQ′ ≥ RQ, the inequality

54

follows from the fact that F ∈ G(F , F

), and the last equality follows from the definition of

ΦQ,F,F .

Claim 4.2. Suppose that Q \ Cf 6= ∅. Then

min(Q,f):Cf=Q′:RQ′≥RQ,F∈G(F ,F )

ΦQ,F,F

(L(CφQ,F,F

))= F

(Q−f

)Proof of claim 4.2.

Observe that, because Q \ Cf 6= ∅ and F (Q) ≤ F (Q) for all Q for all F ∈ G(F , F ),

we must have Q \ Cf 6= ∅ for all F ∈ G(F , F ). This implies that for all Q such that

Cf = Q′ : RQ′ ≥ RQ for some F ∈ G(F , F ), Q− is well-defined. In particular, Q−f

is

well-defined.

By construction, we have L(CφQ,F,F

)= Q− and ΦQ,F,F (Q−) = F (Q−) for all (Q, f)

such that Cf = Q′ : RQ′ ≥ RQ, F ∈ G(F , F ).

Then we need to solve

minQ:Cf=Q′:RQ′≥RQ for some F∈G(F ,F )

F (Q−)

Q−f

minimizes the above expression because F is increasing and

Qf = minQ : F (Q) ≥ BQ= minQ : Cf = Q′ : RQ′ ≥ RQ for some F ∈ G(F , F )

Claim 4.3. Suppose that Q \ Cf 6= ∅. Then

minF∈G(F ,F )

F (L(Cf )) = ΦQf ,F ,F

(L

(CφQ

f,F ,F

))

Proof of claim 4.3.

Observe that the fact that Q \ Cf 6= ∅ implies that L(Cf ) is well-defined for all F ∈G(F , F ). Moreover, it implies that Q−

fis well-defined.

55

We have

minF∈G(F ,F )

F (L(Cf )) = min(Q,f):Cf=Q′:RQ′≥RQ,F∈G(F ,F )

ΦQ,F,F

(L(CφQ

f,F,F

))= F

(Q−f

)= ΦQf ,F ,F

(Q−f

)= ΦQf ,F ,F

(L

(CφQ

f,F ,F

))where the first equality follows from claim 4.1 (note that claim 4.1 applies because Q\Cf 6= ∅for all F ∈ G(F , F )), the second equality follows from claim 4.2, the third equality follows

from the fact that ΦQf ,F ,F

(Q−f

)= F

(Q−f

)and the last inequality follows from the fact

that Q−f

= L

(CφQ

f,F ,F

).

Claim 4.4.

maxF∈G(F ,F )

U(Cf , f) = U

(CφQ

f,F ,F

, φQf ,F ,F

)Proof of claim 4.4.

Suppose first that Q \ Cf = ∅. Then U(Cf , F ) ≥ U(Cf , f) for all F ∈ G(F , F ) and

ΦQf ,F ,F= F , so the claim holds. Suppose next that Q \ Cf 6= ∅. Because this implies that

Q\Cf 6= ∅ for all F ∈ G(F , F ), lemma 12 and claim 4.3 apply. Lemma 12 implies that, given

any F1, F2 ∈ G(F , F ), we have U(Cf1 , f1) > U(Cf2 , f2) if and only if F1(L(Cf1)) < F2(L(Cf2)).Then the result follows from claim 4.3.

Claim 4.5.

minP∈Ω(f)

U(P , f) = U(Cf , f)

Proof of claim 4.5.

Proposition 4 implies that for all F there exists an equilibrium σ such that Cf = Q(σ) ∈Ω(f), U(Q(σ), f) < U(P , f) for all P ∈ Ω(f) such that P 6= Q(σ).

Thenmax

F∈G(F ,F )minP∈Ω(f)

U(P , f) = maxF∈G(F ,F )

U(Cf , f)

= U

(CφQ

f,F ,F

, φQf ,F ,F

)= minP∈Ω

(φQ

f,F ,F

)U(P , φQf ,F ,F

)where the first equality follows from claim 4.5, the second equality follows from claim 4.4,

and the last equality follows from claim 4.5.

Lemma 13. arg min(d,ρ):d∈1,...,m,ρ∈N,ρ≤n−dmde

dρ+1

is given by

1. (2, n− dm2e) if n+ 1 < 2m− dm

2e

56

2. (1, n−m) if n+ 1 > 2m− dm2e

3. (2, n− dm2e), (1, n−m) if n+ 1 = 2m− dm

2e

Proof of lemma 13.

Because ρ 7→ dρ+1

is strictly decreasing, the constraint must bind, so we must have

ρ = n− dmde. Substituting ρ = n− dm

de into d

ρ+1, we obtain d

n−dmde+1

. We define

g(d) =d

n− dmde+ 1

Then the problem is equivalent to solving mind∈1,...,m g(d).

Claim 13.1. d 7→ d(d+1)(d−1)d2−d−1

is strictly increasing for d ≥ 3.

Proof of claim 13.1.

Observe that∂

∂d

[d(d+ 1)(d− 1)

d2 − d− 1

]∝ d4 − 2d3 − 2d2 + 1

and d4 − 2d3 − 2d2 + 1 > 0 is satisfied for d ≥ 3.

Claim 13.2. If⌈md

⌉−⌈

md+1

⌉> 1, then d ≤ m

2.

Proof of claim 13.2.

Suppose for the sake of contradiction that⌈md

⌉−⌈md+1

⌉> 1 and d > m

2. Then we

have md< 2. Because d ≤ m − 1, this implies that m

d∈ (1, 2) and dm

de = 2. Moreover, we

must then have d md+1e ∈ [1, 2], which implies that dm

de − d m

d+1e ≤ 1. This contradicts our

assumption that⌈md

⌉−⌈

md+1

⌉> 1.

Claim 13.3. Suppose that one of the following holds:

1. m = 3, d = 2

2. m = 4 and d = 2 or d = 3

3. d = 2, m > 4

Then n+ 1 > (d+ 1)⌈md

⌉− d⌈

md+1

⌉is satisfied.

Proof of claim 13.3.

Because n ≥ m + 1, we have n + 1 ≥ m + 2. Then it is sufficient to show that

m+ 2 > (d+ 1)⌈md

⌉− d⌈

md+1

⌉.

Consider first case 1. We want to show that 3 + 2 > 3⌈

32

⌉− 2

⌈33

⌉. This is equivalent

to 5 > 4, which is satisfied.

57

Consider next case 2. We want to show that 4+2 > 3⌈

42

⌉−2⌈

43

⌉. This is equivalent to

6 > 2, which is satisfied. We also want to show that 4 + 2 > 4⌈

43

⌉−3⌈

44

⌉. This is equivalent

to 6 > 5, which is satisfied.

Finally, consider case 3. We want to show that m + 2 > 3⌈m2

⌉− 2

⌈m3

⌉. Because

m2

+ 1 >⌈m2

⌉, it is sufficient to show that m + 2 ≥ 3(m

2+ 1) − 2m

3. This is equivalent

to m ≥ 6. Then the only case that is left is d = 2, m = 5. We want to show that

7 > 3⌈

52

⌉− 2

⌈53

⌉. This is equivalent to 7 > 5, which is satisfied.

Claim 13.4. d 7→ g(d) is strictly increasing for d > 1.

Proof of claim 13.4.

We will show that g(d+ 1) > g(d) for all d ∈ 2, . . . ,m− 1.

g(d+ 1) > g(d) is equivalent to d+1

n−d md+1e+1

> d

n−dmd e+1. This simplifies to

n+ 1 > (d+ 1)⌈md

⌉− d⌈ m

d+ 1

⌉(15)

Suppose first that⌈md

⌉−⌈

md+1

⌉≤ 1. If n+ 1 >

⌈md+1

⌉+ d+ 1, we have

n+ 1 >⌈ m

d+ 1

⌉+ d+ 1

≥ (d+ 1)⌈md

⌉− d⌈ m

d+ 1

⌉because

⌈md+1

⌉+ d+ 1 ≥ (d+ 1)

⌈md

⌉− d⌈

md+1

⌉is equivalent to

⌈md

⌉−⌈

md+1

⌉≤ 1.

Then it is sufficient to show that n + 1 >⌈

md+1

⌉+ d + 1, which is equivalent to

n− d >⌈

md+1

⌉.

If n − d ≥ n−1d+1

+ 1, we have n − d > n−1d+1

+ 1 >⌈n−1d+1

⌉≥⌈md+1

⌉where the second

inequality follows from the fact that⌈n−1d+1

⌉< n−1

d+1+ 1 and the third inequality follows from

the fact that m ≤ n− 1.

Then it is sufficient to show that n − d ≥ n−1d+1

+ 1. This is equivalent to n ≥ d + 2.

Because d ≤ m− 1 and m ≤ n− 1, the inequality n ≥ d+ 2 is satisfied.

Suppose next that⌈md

⌉−⌈

md+1

⌉> 1. By claim 13.2, d ≤ m

2holds in this case.

58

If n+ 1 ≥ (d+ 1)(md

+ 1)− d m

d+1, we have

n+ 1 ≥ (d+ 1)(md

+ 1)− d m

d+ 1

> (d+ 1)⌈md

⌉− d

⌈m

d+ 1

⌉where the second inequality follows from the fact that dm

de < m

d+ 1 and d m

d+1e ≥ m

d+1.

Then it is sufficient to show that n + 1 ≥ (d + 1)(md

+ 1)− d m

d+1, which is equivalent

to n ≥ m(1 + 1

d− d

d+1

)+ d.

If m + 1 ≥ m(1 + 1

d− d

d+1

)+ d, we have n ≥ m + 1 ≥ m

(1 + 1

d− d

d+1

)+ d where

the first inequality follows from the fact that m ≤ n − 1. Then it is sufficient to show that

m+1 ≥ m(1 + 1

d− d

d+1

)+d, which is equivalent to m(d2−d−1) ≥ (d+1)(d−1)d. Because

d2 − d− 1 > 0 for d > 1, this is equivalent to m ≥ d(d+1)(d−1)d2−d−1

.

Suppose first that d ≥ 3 and m ≥ 5. Because d ≤ m2

and, by claim 13.1, d 7→ d(d+1)(d−1)d2−d−1

is strictly increasing for d ≥ 3, it is sufficient to show that m ≥m2

(m2

+1)(m2−1)

(m2

)2−m2−1

. This is

equivalent to m2 − 4m− 4 ≥ 0, which holds for m ≥ 5.

Suppose next that either d ≥ 3 or m ≥ 5 fails. Because we have m ≥ 2 and d > 1,

d ≤ m− 1, the cases covered in claim 13.3 exhaust the universe of the possible cases. Then

claim 13.3 implies that (15) is satisfied, as required.

Claim 13.4 implies that mind∈1,...,m g(d) = ming(1), g(2). g(2) > g(1) is equivalent

to 2

n−dm2 e+1> 1

n−m+1, which simplifies to n+ 1 > 2m− dm

2e.

Lemma 14. Fix P ⊂ Q satisfying P 6= ∅. Then for all κ ∈(0, m

n

)there exists a distribution

of proposal power f such that U(P , f) = κ.

Proof of lemma 14.

Given ε ∈ (0, 1), let f(Q) = ε|P| for all Q ∈ P and let f(Q) = 1−ε

|Q\P| for all Q ∈ Q \ P .

Because P 6= ∅ and Q \ P 6= ∅, this is well-defined. Then we have U(P , f) = mn

ε1−δ+δε .

Because ε 7→ mn

ε1−δ+δε is continuous, m

nε

1−δ+δε |ε=0= 0 and mn

ε1−δ+δε |ε=1= m

n, for all κ ∈

(0, m

n

)there exists ε ∈ (0, 1) such that m

nε

1−δ+δε = κ.

Lemma 15. mn>

DQρQ+1

for all Q 6= m.

Proof of lemma 15.

We will show that mn> maxQ∈Q\m:d=DQ

dρQ+1

. Note that, given that DQ is d, the

maximum number of proposers is |Q| = 1 +m− d (which is achieved under Q in which one

agent is a proposer on d pies and there is exactly one proposer on each of the remaining

59

pies). Because ρQ = n− |Q|, this implies that minQ∈Q\m:d=DQ ρQ = n− 1−m + d. Then

we have maxQ∈Q\m:d=DQd

ρQ+1= d

n−m+d. Because d 7→ d

n−m+dis strictly increasing, we have

m−1n−1≥ DQ

ρQ+1for all Q 6= m. Moreover, m

n> m−1

n−1implies that m

n>

DQρQ+1

for all Q 6= m,as required.

Lemma 16. Suppose that q > m. ThenDQsQ+1

is maximized at Q = m and is minimized

at Q = D1 if q + 1 > 2m− dm2e and at Q = D2 if q + 1 < 2m− dm

2e, where D1 and D2 are

as defined in section C of the Appendix.

Proof of lemma 16.

Observe that the indexDQsQ+1

when the number of agents is n and the voting rule is

q > m is equal to the indexDQρQ+1

when the number of agents q > m and the voting rule

is unanimity. This is because if q > m, then q > |Q| for all Q, so that sQ = q − |Q| and

ρQ = n − |Q|. Then the result thatDQsQ+1

is maximized at Q = m follows from lemma 15

and the result thatDQsQ+1

is minimized at Q = D1 if q + 1 > 2m − dm2e and is minimized at

Q = D2 if q + 1 < 2m− dm2e follows from lemma 13.

Proof of Theorem 4.

Let Mnκ =

Q :

DnQsQ,n+1

≥ κ

. Suppose first that limn→∞∑

Q∈Qnγfn(Q) = 0 for all γ > 0.

Claim 4.6. Q ∈ limn→∞Mnκ for some κ > 0 if and only if Q ∈ limn→∞Q

nγ for some γ > 0.

Proof of claim 4.6.

Observe thatsQ,nn≥ qn

n−mn

n. This implies that limn→∞ sQ,n ≥ βn for β = qn

n−mn

n. The

fact that limn→∞mnn< limn→∞

qnn

implies that β > 0. Then limn→∞DnQ

sQ,n+1= limn→∞

DnQsQ,n≤

limn→∞1β

DnQn

implies that if limn→∞DnQ

sQ,n+1≥ κ, then limn→∞

DnQn≥ γ = κβ > 0. Conversely,

becauseDnQ

sQ,n+1≥ DnQ

n, if limn→∞

DnQn≥ γ for some γ > 0, then limn→∞

DnQsQ,n+1

≥ κ for some

κ > 0.

Claim 4.7. If limn→∞∑

Q∈Cfnfn(Q) = 0, then limn→∞minQ∈Cfn

DnQsQ,n+1

= 0. If

limn→∞∑

Q∈Cfnfn(Q) > 0, then limn→∞minQ∈Cfn

DnQsQ,n+1

> 0.

Proof of claim 4.7.

Let L = limn→∞∑

Q∈Cfnfn(Q). Suppose for the sake of contradiction that L = 0 and

limn→∞minQ∈CfnDnQ

sQ,n+1> 0. Then there exists Q 6∈ Cfn such that

DnQsQ,n+1

> 0. Because

Q 6∈ Cfn , Theorem 1 implies that δV σn >DnQ

sQ,n+1> 0. Lemma 7 implies that limn→∞ V

σn =

limn→∞mnn

L1−δ+δL . Then limn→∞ δ

mnn

L1−δ+δL > 0, which contradicts L = 0.

Next, suppose that L > 0. Theorem 1 implies that δV σn ≤ DnQsQ,n+1

for all Q ∈ Cfn .

Because limn→∞ Vσn = limn→∞

mnn

L1−δ+δL , we have limn→∞

DnQsQ,n+1

≥ δ limn→∞mnn

L1−δ+δL > 0

60

for all Q ∈ Cfn because L > 0.

Claim 4.8. If limn→∞∑

Q∈Qnγfn(Q) = 0 for all γ > 0, then limn→∞minQ∈Cfn

DnQsQ,n+1

= 0. If

limn→∞∑

Q∈Qnγ′fn(Q) > 0 for some γ′ > 0, then limn→∞minQ∈Cfn

DnQsQ,n+1

> 0.

Proof of claim 4.8.

Suppose that limn→∞∑

Q∈Qnγfn(Q) = 0 for all γ > 0. Claim 4.6 then implies that

limn→∞∑

Q∈Mnκfn(Q) = 0 for all κ > 0. Note that Cfn = Mn

κ for some κ > 0. Then

limn→∞∑

Q∈Mnκfn(Q) = 0 for all κ > 0 implies that limn→∞

∑Q∈Cfn

fn(Q) = 0. Claim 4.7

implies that if limn→∞∑

Q∈Cfnfn(Q) = 0, then limn→∞minQ∈Cfn

DnQsQ,n+1

= 0.

We now show that if limn→∞∑

Q∈Mnκfn(Q) > 0 for some κ > 0, then

limn→∞∑

Q∈Cfnfn(Q) > 0. Suppose not, so that limn→∞

∑Q∈Cfn

fn(Q) = 0. Then

claim 4.7 implies that limn→∞minQ∈CfnDnQ

sQ,n+1= 0, so that limn→∞ Cfn = limn→∞M

n0 .

Because limn→∞∑

Q∈Mnκfn(Q) > 0, we have limn→∞

∑Q∈Mn

0fn(Q) > 0. Then 0 =

limn→∞∑

Q∈Cfnfn(Q) = limn→∞

∑Q∈Mn

0fn(Q) > 0, which is a contradiction.

Next, suppose that limn→∞∑

Q∈Qnγ′fn(Q) > 0 for some γ′ > 0. Claim 4.6 implies

that limn→∞∑

Q∈Mnκfn(Q) > 0 for some κ > 0. The argument above implies that

limn→∞∑

Q∈Cfnfn(Q) > 0. Claim 4.7 then implies that limn→∞minQ∈Cfn

DnQsQ,n+1

> 0.

Fix Q such that RQ ≤ RQ′ for all Q ∈ Cfn and RQ ≥ RQ′′ for all Q′′ 6∈ Cfn . Let

κn =sQ,n+1

n. Because Q ∈ Cfn , Theorem 1 implies that (sQ,n + 1)δV σn ≤ Dn

Q. Because

(sQ,n + 1)V σn = κnnVσn , this is equivalent to κnnδV

σn ≤ DnQ. That is, κnδV

σn ≤ DnQn

.

We want to show that limn→∞nmnV σn = 0. Claim 4.8 implies that 0 =

limn→∞minQ∈CfnDnQ

sQ,n+1≥ limn→∞

DnQn

. Because κnδVσn ≤ DnQ

nand limn→∞

DnQn

= 0 imply

that limn→∞ κnδVσn ≤ limn→∞

DnQn

= 0, it is enough to show that nmn

< αδκn for some

α ∈ (0,∞). Note that sQ,n = qn − |Q| ≥ qn −mn implies that κn ≥ sQ,nn≥ qn−mn

n. Then it

is enough to show that nmn

< αδ qn−mnn

for some α ∈ (0,∞).

Let ε = limn→∞qn−mn

nand let η = limn→∞

nmn

, so that limn→∞nm

nq−m = η

ε. Let α = 2η

ε1δ.

Note that α < ∞ because ε > 0. nmn

< αδ qn−mnn

is equivalent to limn→∞nm

nqn−mn

1δ< α.

Because, by construction, limn→∞nm

nqn−mn = δ α

2, this is satisfied.

Next suppose that limn→∞∑

Q∈Qnγfn(Q) > 0 for some γ > 0. We want to show that

limn→∞nmnV σn > 0. Claim 4.8 implies that limn→∞

DnQsQ,n+1

> 0 for all Q ∈ limn→∞ Cfn . Then

limn→∞maxQ 6∈CfnDnQ

sQ,n+1> 0. Because δV σn ≥ DnQ

sQ,n+1for all Q 6∈ Cfn by Theorem 1, this

implies that limn→∞ Vσn > 0. Because n

mn> 1 for all n, we then have limn→∞

nmnV σn > 0,

as required.

Proposition 5. Suppose that m = 4, n = 6 and δ = 910

. ThenD1,1,1,1ρ1,1,1,1+1

= 13,

D2,2ρ2,2+1

= 25,

61

D1,1,2ρ1,1,2+1

= 12,

D1,3ρ1,3+1

= 35,

D4ρ4+1

= 23. Moreover, Cf = 2, 2, 1, 1, 2, 1, 3, 4 and

U(Cf , f) = 59.

Proof of Proposition 5.

Observe that D1,1,1,1 = 1, ρ1,1,1,1 = n − 4, D2,2 = 2, ρ2,2 = n − 2, D1,1,2 = 2,

ρ1,1,2 = n − 3, D1,3 = 3, ρ1,3 = n − 2, D4 = 4, ρ4 = n − 1. Since n = 6, we have

ρ1,1,1,1 = 2, ρ2,2 = 4, ρ1,1,2 = 3, ρ1,3 = 4, ρ4 = 5.

ThenD1,1,1,1ρ1,1,1,1+1

= 1n−3

,D2,2ρ2,2+1

= 2n−1

,D1,1,2ρ1,1,2+1

= 2n−2

,D1,3ρ1,3+1

= 3n−1

,D4ρ4+1

= 4n. Since

n = 6, we haveD1,1,1,1ρ1,1,1,1+1

= 13,

D2,2ρ2,2+1

= 25,

D1,1,2ρ1,1,2+1

= 12,

D1,3ρ1,3+1

= 35,

D4ρ4+1

= 23.

It can be shown that Cf = 2, 2, 1, 1, 2, 1, 3, 4. Then∑

Q∈Cf f(Q) = 1 −

f(1, 1, 1, 1) = 1 − 1624

= 13

and U(Cf , f) = mn

∑Q∈Cf

f(Q)

1−δ+δ∑Q∈Cf

f(Q)= 2

3

13

1−δ+δ 13

= 23

13−2δ

. Because

δ = 910

, we have U(Cf , f) = 59.

Example: Order on the Proposing Partitions Varies with the Voting Rule

There are 3 pies and 4 agents. The proposing partitions are 3, 1, 2 and 1, 1, 1.We make the dependence of sQ on q explicit by writing sqQ. If q = 4, then s4

3 = 3,

s41,2 = 2 and s4

1,1,1 = 1, so thatD3s43+1

= 34,

D1,2s41,2+1

= 23

andD1,1,1s41,1,1+1

= 12. Therefore,

R3 > R1,2 > R1,1,1 under q = 4.

If q = 3, then s33 = 2, s3

1,2 = 1 and s31,1,1 = 0, so that

D3s33+1

= 1,D1,2s31,2+1

= 1 and

D1,1,1s31,1,1+1

= 1. Therefore, R3 = R1,2 = R1,1,1 under q = 3.

If q = 2, then s23 = 1, s3

1,2 = 0 and s31,1,1 = 0, so that

D3s23+1

= 32,

D1,2s21,2+1

= 2 and

D1,1,1s31,1,1+1

= 1. Therefore, R1,2 > R3 > R1,1,1 under q = 2.

Lemma 17. Suppose that q > m. Then if DQ1 ≥ DQ2, RQ1 ≥ RQ2 under q = q1 and

q2 ≥ q1, we have RQ1 ≥ RQ2 under q = q2.

Proof of lemma 17.

We make the dependence of sQ on q explicit by writing sqQ.DQ1

sq1Q1

+1≥ DQ2

sq1Q2

+1is equivalent

to DQ1sq1Q2−DQ2s

q1Q1≥ DQ2 −DQ1 and

DQ1

sq2Q1

+1≥ DQ2

sq2Q2

+1is equivalent to DQ1s

q2Q2−DQ2s

q2Q1≥

DQ2 −DQ1 . Then it is enough to show that DQ1sq2Q2−DQ2s

q2Q1≥ DQ1s

q1Q2−DQ2s

q1Q1

. This is

equivalent to DQ1(sq2Q2− sq1Q2

) ≥ DQ2(sq2Q1− sq1Q1

).

62

Because DQ1 ≥ DQ2 and sq2Q2− sq1Q2

≥ 0, it is enough to show that

DQ2

(sq2Q2− sq1Q2

)≥ DQ2

(sq2Q1− sq1Q1

)(16)

Thus we want to show that sq2Q2−sq1Q2

≥ sq2Q1−sq1Q1

. Because q > m, we have sqQ = q−|Q|for q = q1, q2 and Q = Q1, Q2. Then (16) is equivalent to q2− |Q2| − q1 + |Q2| ≥ q2− |Q1| −q1 + |Q1|, which is satisfied.

63

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