behavioral mechanism design

8/10/2019 Behavioral Mechanism Design

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BEHAVIORALMECHANISMDESIGN

SERKAN KUCUKSENEL

Middle East Technical University

Abstract

This paper studies the mechanism design problem for theclass of Bayesian environments where agents do care forthe well-being of others. For these environments, we fully

characterize interim efficient (IE) mechanisms and exam-ine their properties. This set of mechanisms is compelling,since IE mechanisms are the best in the sense that there isno other mechanism which generates unanimous improve-ment. For public good environments, we show that thesemechanisms produce public goods closer to the efficientlevel of production as the degree of altruism in the pref-erences increases. For private good environments, we showthat altruistic agents trade more often than selfish agents.

1. Introduction

A group of individuals must choose an alternative from the set of possible al-ternatives and must decide how to arrange monetary transfers. Initially, eachagent has private information about each possible alternative. An agentsutility for a given alternative depends not only on her own material utilitybut also on the welfare of other agents. This implies agents are unselfish oraltruistic. In this framework, we characterize the most efficient mechanisms

within the mechanisms that satisfy incentive and feasibility constraints.The assumption of self-interest is problematic. The self-interest hypothe-

sis states that preferences among allocations depend only on an agents ownmaterial well-being. Experimental results suggest that people often do care

Serkan Kucuksenel, Department of Economics, Middle East Technical University, Ankara,06800, Turkey ([email protected]).

I thank John Ledyard, Thomas Palfrey, Federico Echenique, and Antonio Rangel for

their help and encouragement. I also wish to thank seminar participants at Caltech,METU, and Sabanci University for valuable comments and suggestions on an earlier ver-sion of this paper.

Received July 7, 2010; Accepted January 31, 2011.

C2012 Wiley Periodicals, Inc.Journal of Public Economic Theory, 14 (5), 2012, pp. 767789.

767


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Behavioral Mechanism Design 769

trade. Moreover, altruistic agents do not trade when it is not optimal to trade.We also show that probability of efficient trade converges to one when fullsocial preferences are in action.

The remainder of the paper is organized as follows. In the next section,we describe the environment and introduce the basic notation. In Section 3we present the characterization results. Section 4 provides applications ofour characterization for both public good and private good environments.Finally, we summarize the findings of the paper and make some concludingremarks in Section 5. The proofs are relegated to the Appendix.

2. The Model

Consider a Bayesian mechanism design framework with n agents. The setof agents is denoted byN = {1, . . . , n}. Each agent has a type i which isher private information. We assume that each agent knows her own typeand does not know the types of the other agents. Each i is indepen-

dently drawn from cumulative distribution functionFi(.) on i = [i, i]

with 0 i i

< . Types are drawn independently across agents; that is,the is are independent random variables. We denote a generic profile ofagent types by = (1, . . . , n) Ni=1

i. For any , we adopt thestandard notation so thati = (1, . . . , i1, i+1, . . . , n), and = (i, i)

where f() = Ni=1 fi(i). LetXbe a finite set of possible nonmonetary de-cisions, or allocations (e.g.,Xcould be a subset of an Euclidean space andrepresent the set of possible allocations of private and public goods).

Let (X) be the set of probability distributions on X. A mechanism = (y, t) consists of an allocation rule yand a payment rule t. Let yx()denote the probability of choosingx X, given the profile of types . Afeasible allocation rule (or social choice function) y : (X) is a func-tion from agents reported types to a probability distribution over allocationssuch thatxX y

x() = 1 and yx() 0 for all . We allow allocation

rules to randomize over feasible allocations. LetYbe the set of all possibleallocation rules and Y be the set of all feasible allocation rules. Thepayment rulet : RN is a map from the agents reported types to mon-

etary compensations whereN

i=1ti() 0. This condition (ex post budget

balance) requires that there is no outside source to finance the compensa-tions. Therefore, a mechanism cannot run a deficit.

The individual payoff function (or material utility) of an agent igivenan allocation ruley, and her monetary paymentti is

i(y, ti, i) = xX

yx()vi(x, i) ti,

wherevi(x, i) is agentis valuation of allocation xwhich depends on herprivate information. We assume that vi(x, i) is differentiable, monotoneincreasing, and convex in i for alliandx X.


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770 Journal of Public Economic Theory

Beyond her individual payoff, agenticares about the payoffs of others:

ui(y, t, ) = ii + (1 i)

= xX

yx()Vi(x, , i) + ijN tjN

ti jNtjN

,

where =

jNj

N is the average payoff in the population andVi(x, , i) =

ivi(x, i) + (1 i)

jNvj(x,j)

N is the total value of allocationxfor agenti.

The constanti [0, 1] is an agent-specific weighting factor showing eachagents social concerns. Ifi = 1, the agent has selfish preferences which donot directly depend on the well-being of others. Ifi 1).

We only consider direct mechanisms in which the set of reported typesis equal to the set of possible types in the rest of the paper. By the revelationprinciple, any allocation rule that results from equilibrium in any mecha-nism is also an equilibrium allocation rule of an incentive compatible, directmechanism. Therefore, there is no loss of generality in restricting our atten-tion to these simple type of mechanisms.

LetUi(, i, si) be the interim expected utility of agent iwhen he re-portssi = i, assuming all other agents truthfully report their type. That is

Ui

(, i

, si

) = Ei[ui

(y(si

, i

), t(si

, i

), )].

DenoteUi(, i) Ui(, i, i). Theex anteutility of agentiis

Ui() = E[ui(y(), t(), )].

Define also the conditional expected payment function ai :i R suchthat

ai(i) = Ei[ti()].

A mechanism is interim incentive compatible (IIC) if honest reportingof types defines a BayesianNash equilibrium. That is, is IIC if and onlyifUi(, i) Ui(, i, si) for all i, si, i. We call a mechanism interim indi-

vidual rational (IIR) if every agent wants to participate in the mechanism:Ui(, i) 0 for alli, i. A mechanism isex antebudget balanced (EABB) ifa mechanism designer does not expect to pay subsidies to the agents, e.g.,


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B() = E(N

i=1ti()) 0. A mechanism is feasible if it satisfies IIC, IIR,

and EABB.A mechanismis IE if it is feasible and there is no other feasible mech-

anismsuch thatUi( , i) Ui(, i) for alli, i andUi( , i)>Ui(, i)for someiand for all i i i, wherei has a strictly positive measurerelative to i. IE is an extension of efficiency to the environments with pri-

vate information. A mechanism is IE if there does not exist an alternativefeasible mechanism that interim Pareto-dominates it. Note that the idea ofPareto-domination is applied to the expected utilities after the agents havelearned their types. IE mechanisms can also be represented as the solutionsto a set of maximization problems. A mechanism is an IE mechanism if

and only if there exists = {i : i R+}Ni=1with

i

i i(i)dFi (i)>0 for

somei, such thatmaximizesNi=1 ii i(i)Ui(, i)dFi (i) subject to isfeasible. Note that the weight attached to an agentican vary with her type.1

Thus, an IE mechanism maximizes weighted sum of agents utilities subjectto IIC, IIR, and EABB constraints.2

3. IE Mechanisms: The Characterization

We now proceed to characterize the set of IE mechanisms. We first refor-

mulate the constraints set such that we can provide necessary and sufficientconditions for IIC and IIR constraints. The second step in the characteriza-tion involves a general solution to the maximization problem with the con-straints rewritten. Following the same idea in Myerson (1981), we find thesolution to the case where the second-order IIC condition is not binding(regular problems). Then, we provide a sufficient condition in which thesolution to the regular problem coincides with the solution to the originalproblem.

Welfare weights play an important role in our analysis. Before stating the

main characterization, the following definition will be useful in reformulat-ing the original problem.

DEFINITION1: If 0i i

i i(i)dFi(i)> 0, leti(i) = 1

0i

ii

i(s)dFi(s).

If 0i = 0,leti(i) = 0.

Let0i denote agent isex antewelfare weight relative to other agents.i(i) is a relative weight of agent is lower types given her private

information.

1 See Holmstrom and Myerson (1983) for more details on this.2We could also useex postbudget balance condition. It turns out that EABB is equivalenttoex postbudget balance condition in our setting.


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We can now provide our first characterization. This result implies thatthe objective function can just be written as a function of utilities of thelowest typesUi(, i) and the allocation ruley. The objective function does

not depend on the transfers anymore.

THEOREM 1: A mechanism = (y, a) is IE if and only if there exists nonnega-tive type-dependent welfare weights, {i}Ni=1, where

iN

0i >0, and N constants,{ai(i)}Ni=1, such that(y, {a

i(i)}Ni=1)solves,

maxy

Ni=1

0i

Ui(, i) +

ii

1 i(i)

fi(i)

Qi(i, )dFi (i)

(1)

subject to3

B() = (, ) N

i=1

Ui(, i) 0, (2)

Ui(, i) =

i

xX

yx(i, i)Vi(x, i, i, )dFi(i)

ai(i) 0

(3)

Qi(i, )monotone increasing for all i, i. (4)

Given nonnegative type-dependent welfare weights, {i}Ni=1, we can de-fine the Lagrangian function as

Ly, (ai(i))Ni=1, , (

i)Ni=1

=

Ni=1

0i

Ui(, i) +

ii

1 i(i)

fi(i)

Qi(i, )dFi (i)

+ (, ) Ni=1

Ui(, i)+

Ni=1

i

i

xX

yx(i, i)Vi(x, i, i, )dFi(i) ai(i)

.

The first-order conditions with respect to (EABB multiplier) and with re-spect to i (IIR multiplier) imply that

0, B() 0 and B() = 0,

i 0, Ui(, i) 0 and iUi(, i) = 0 for alli N.

3 The expected budget surplus in an IIC mechanism is equal to B(), where (, ) Ni=1

xX y

x()(Vi(x, , ) Vi(x,,)

i1Fi(i)

fi(i) )dF().


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The first-order condition with respect to ai(i) yields 0i + i = 0.Then 0i for alli N. This implies the EABB constraint is always bind-ing ( >0) since there is i N such that0i >0 and i 0 for all i N.

The intuition for this result is the following. We assumed that contributionsin excess are not socially valued. If the EABB is not binding, a redistribu-tion of budget surplus to the agents would result in an interim Pareto im-provement. If the IIR constraints for the lowest types of agents with nonmax-imal expected welfare weight are not binding, a redistribution of wealth toagents with maximal expected welfare weight would increase the value of the

weighted welfare function.When we substitute and rearrange terms in our original problem, we

get the following result. This result is a simplified reformulation of Theorem

1 for regular problems. The utilities of lowest types Ui

(, i

) are explicitlyentered into the objective function and the constraints of the maximizationproblem reduce to two constraints representing the EABB constraint.

THEOREM 2: For regular problems, there is a payment rule{ai}Ni=1such that =(y, a) is IE if and only if there exists nonnegative type-dependent welfare weights,{i}Ni=1 with

iN

0i >0, and such that y (X) simultaneously solvesthe following inequalities:

maxy

N

i=1 xXyx()

Vi(x, , ) +

Vi(x, , )

i

Fi(i) 1

fi(i) +

0i

1 i(i)

fi(i)

dF(), (5)

0 (, ), (6)

0 = ( )(, ). (7)

3.1. Modified Virtual Valuations

In this section we show the effects of interdependent preferences in our set-ting. Let

Wi(x, , , i) = Vi(x, , )

+Vi(x, , )

i

Fi(i) 1

fi(i) +

0i

1 i(i)

fi(i)

. (8)

We callW

i

(x, , , i

) as the (modified) virtual valuation of agentifor allo-cation xfollowing Myerson (1981). Rather than directly working with totalvaluations, IE mechanisms use the agents total valuations suitably adjusted.The virtual valuation for a given allocation is equal to the agents total valua-tion for the allocation,Vi(x, , ), with two adjustments that depend on thedistribution of types, welfare weights, and social concerns of the agents. The


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first one, Vi(x,,)

iFi(i)1

fi(i) , is due to the informational rent to be given for

truthful revelation of the agents private information. The second one is due

to distortions arising from redistribution of income ( Vi(x,,)

i0i

1i(i)fi(i)

).

Note that these adjustments are weighted with the marginal total valuation ofagentifor a given allocation and hence virtual valuations also depend on theallocation.

3.2. A Sufficient Condition for Regularity

The problem stated in Theorem 2 has a simple solution defined by4

yx

(, ) = 1 ifx= arg maxmXNi=1W

i(m, , , i)

0 otherwise. (9)

This implies an IE mechanism assigns probability one to an allocation withthe highest sum of modified virtual valuations. Note that to find the IE mech-anism we use the minimum possible such that (6) and (7) are satisfied.

We now provide a condition under which the solution (9) and the con-dition imply that the monotonicity constraint is satisfied.

ASSUMPTION 1: (a) W i(x, , , i) is nondecreasing inifor all i N,x X,

and all ,and (b) Vi(x,,)i

is nondecreasing in x for all i N,all ,andall x X.

Note that we did not make any assumption about how the total valu-ations depend on the allocation up to now. Assumption 1 basically restrictsthe set of admissible valuation functions and welfare weights such that the so-lution to the reduced problem satisfies the monotonicity condition. Assump-tion 1(a) reduces to a joint condition on the priors, welfare weights, and thecurvature of the total valuation functions. We already know by the initial as-

sumption that Vi(x,i,)

i increases (decreases) when i increases (decreases).

However, the allocation might also change as a result of an increase in anagents signal. Note that the allocation cannot decrease due to Assumption1(a). Assumption 1(b) guarantees that the derivative of the total valuation

functions Vi(x,,)

i are also nondecreasing in x. For example, if priors are

uniform on [0, 1],Vi(x, , ) =x(i + (1 )

jNj

N ), and = 0 then

W

i

()

= jNj

N+

1

N i 1 + 0i

(1 i

(i

) .4 For simplicity, we assume that there are no allocations x,y, x = y such thatN

i=1Wi(x, , , i) =

Ni=1W

i(y, , , i). We could also use a random tie-breaking rule.


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So Assumption 1(a) requires Wi

i 0. This is true if and only ifi(i) 2

for all i N and all i i. We know that and welfare weights arealways nonnegative. Therefore, this condition is satisfied for all possible wel-

fare weights. For general priors and social concerns, this assumption requires

i(i)

2fi(i) fi(i)

i (Fi(i)i(i))

fi(i)

.

This implies Assumption 1(a) may not be satisfied for all welfare weightswith arbitrary priors. We showed that with uniform priors the assumptioncan be satisfied for all welfare weights and thus it is satisfied by all incentiveefficient mechanisms. Other than the uniform distribution, the following

set of distributions will satisfy this condition for all welfare weights: for alli N,Fi(i) = (i) where >1 and i [0, 1]. Note also that the totalvaluation function satisfies Assumption 1(b).

THEOREM 3: If each W i() and Vi()satisfies Assumption1,then the solution(9) satisfies all constraints in Theorem 1.

4. Applications

Our characterization is general and can be applied to different economicsettings. In this section, we present the main intuition of the characteriza-tion by providing simplified applications for both public and private goodsenvironments.

4.1. Public Goods

There are Npeople who must decide on the level of a public good whichis produced according to constant returns to scale. In addition, they must

decide how to distribute the production costs. Let X = {0, 1} denote thepossible values of the public good. The cost of producing the public goodis equal to K. In our main model, we assumed that the social allocation iscostless but it is easy to incorporate the cost of a social allocation to ourmodel.

For this application we assume that the total valuation functions have thefollowing form:

Vi(x, , ) =xi + (1 )jN

j

N . (10)This implies the utility function of agenti is

ui(y, t, ) =xX

yx()Vi(x, , ) +

jNt

j

N ti

jNt

j

N. (11)


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If = 1, the model is equivalent to the standard public good environmentwhere agents are selfish. If = 0, the model is a common values settingwhere full social preferences are in action and the society is homogeneous

(every agent has the same valuation for the public good). If 1> 0, agentshave interdependent preferences. Note that as decreases the degree of al-truism in preferences goes up and the model converges to the full socialpreferences setting.

For the regular case, given welfare weights i : i R+, an IE mecha-nism satisfies:

maxy

Ni=1

xX

yx()

Wi(x, , , i)

K

Nx

dF(), (12)

0 (, )

KxX

yx()xdF(), (13)

0 = ( )

(, )

KxX

yx()xdF()

. (14)

Without IIR constraints:Suppose the IIR was not required. It is much eas-

ier to solve the problem without IIR constraints. First-order conditions im-ply 0i = = for all i N. Hence the ex ante welfare weights must allbe equal. Otherwise, the solution does not exist, since it is always possi-ble to improve welfare by making arbitrarily large transfers among agents

with different welfare weights. The following proposition summarizes thisobservation.

PROPOSITION 1: If there is an interim efficient mechanism (without individualrationality), then there exist nonnegative type-dependent welfare weights, {i}Ni=1,suchthat0i = 0jfor all i, j N.

The problem stated above for the regular case has a simple solution:

yx(, ) =

1 ifx= arg maxmX

Ni=1W

i(m, , , i) K m

0 otherwise,(15)

and the payment functions can be found using the formula in the proof ofTheorem 2 after subtracting the expected cost of the public good from the

constraint on the sum of the expected payment of the lowest types of eachagent. The public good is produced if

Ni=1

i +

+

1

N

Ni=1

Fi(i) i(i)

fi(i) K.


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The first best decision is to produce the public good whenN

i=1Vi(1, , ) =

Ni=1V

i(1, , = 1) =

Ni=1v

i(1, ) =

Ni=1

i K. Lete = {|

Ni=1

i

K} and = {|Ni=1W

i() K}. Efficiency dictates that the public good

provision does not depend on the social concerns of agents. In IE mecha-nisms, there are distortions from the first best due to informational rents,the type-dependent welfare weights, and the measure of social concerns.Note that even though the sum of the valuations for the public good isindependent of the social concerns of agents, IE production decisions de-pend on the interdependence among preferences. It is also easy to see that ifi(i) = ci R+for alliand

i,ex anteefficient production decisions whichare also IE correspond to the classical first best decisions (orex postefficientallocations) sinceFi(i) = i(i) for this case.

PROPOSITION 2: (In IE mechanisms without IIR) If the welfare weights are con-stant, i(i) = ci R+ for all i and

i,then the public good is produced if it is expost efficient to produce it.

Now suppose thati(i) is decreasing for all i and i (lower types areweighted more heavily). This implies the aim of the planner is that agentsvaluing the public good more should bear a larger share of the cost. Then,

Ni=1

Fi(i)i(i)fi(i)

0. This implies that there is moreproduction than theex postefficient level for all since the sum of the modi-fied virtual valuations is more than the sum of the valuations. Thus, over pro-duction is a more efficient way to relax incentive compatibility constraintsthan transfers. For this case, welfare is increased either by shifting taxes fromhigh types to low types or by producing the public good more often. If the


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planner wants to shift the tax burden from higher types to lower types with-out violating the IIC constraints, the only possible way is to produce the pub-lic good less often. This would make higher types worse off. Therefore, the

only possible way to increase welfare without violating the constraints is toproduce the public good more often. Moreover, as decreases, the publicgood is produced relatively less often. The following comparative statisticsresults directly follows from the above discussions.

PROPOSITION 3: (In IE mechanisms without IIR) If the welfare weights are de-creasing in type, the public good is produced more often as the degree of altruism inpreferences goes up. If the welfare weights are increasing in type, the public good isproduced less often as the degree of altruism in preferences goes up.

With IIR constraints: Now suppose that the IIR constraints are required.The main question is then whether the individual rationality constraints

will be binding or not for the agent who is assigned the highest welfareweight. We know that individual rationality constraints will be binding for allother agents since . Suppose > . This implies individual rationalityconstraints are binding for all agents. For this case, virtual valuations areequivalent to

Wi

( , , i

) = i + (1 )jNj

N

+

+

1

N

Fi(i) 1

fi(i) +

0i

1 i(i)

fi(i)

.

The public good is produced if

N

i=1i +

+

1

N

N

i=1Fi(i) 1

fi(i) +

0i

1 i(i)

fi(i)

K. (16)

With IIR constraints, virtual valuations are always lower for all agents. Hencethe frequency of IE public good production is always lower with the con-straints than without. This implies that in some cases it might be efficient toproduce the public good but there might not be enough surplus to cover theincentive costs without violating individual rationality constraints. Note that

the adjustment term, ( + 1N

)(N

i=1Fi(i)1

fi(i) +

0i

1i(i)fi(i)

), is always nega-

tive. If IE production occurs, then

Ni=1

i >K. IE mechanisms do not pro-

duce the public good when it is not optimal to produce the public good andthese mechanisms may not produce the public good when it is optimal toproduce. However, the adjustment term becomes smaller as the degree ofaltruism in the preferences goes up. This implies agents earn less informa-tional rents, the budget balance constraint is relaxed and the constrained


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optimum is getting more efficient. It is easier to satisfy individual rationalityconstraints with relatively more unselfish agents. That is, public good provi-sion increases as decreases, and there will be fewer information states of

the economy in which it is optimal to produce the public good but the pub-lic good is not produced. Formally, e

for all , [0, 1] suchthat > . Note that, inefficiency in public good production is the smallest

when full social preferences are in action. The economic intuition behindthis result is the following. If agents care more about others, there will beless need to use transfers and to distort the outcome away from efficiencyin order to satisfy the IIR constraints. Thus, the incentive efficient outcomes

will be closer to ex postefficient outcomes. These observations lead to thefollowing result.

PROPOSITION 4: Interim efficient public good provision increases with the degreeof altruism in preferences.

4.2. Bargaining: One Buyer and One Seller

There is a risk-neutral seller who wants to sell an indivisible object that sheowns and a risk-neutral buyer who wants to buy the object. The sellers type is

s [s, s], and the buyers type isb [b,

b]. We assume that [s,

s]

[b

,

b

] = . That is, there are gains from trade for some information statesof the economy. A nonmonetary decision may be represented by a vectorx=(xs, xb), wherexs = 1 if the seller keeps the good,xs = 0 if the seller sells thegood,xb = 1 if the buyer gets the good, and xb = 0 if the buyer does not getthe good. The set of possible allocations is then X = {(1, 0), (0, 1)}. Agentis individual payoff depends on the decision ruley, her private informationi and her monetary transferti,

i(y, ti, i) =

xXyx()vi(xi, i) ti.

Beyond her individual payoff, agenti {b, s}cares about the payoff ofthe other agent,

ui(y, t, ) =xX

yx()Vi(x, , ) ti (1 )ts + tb

2 .

For this application we assume that total valuation functions have the follow-ing form5:

Vi(x, , ) =

xii + (1 )

xss + xbb

2 .

5 If = 1, the model is equivalent to the original Myerson and Satterthwaite (1983) bar-gaining problem with selfish agents.


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If the parties do not reach an agreement, they get their outside

options. The sellers outside option is U0s(s) = 1+2

s and U0b(b) =

s1

2 sdFs(s) for the buyer, the buyers expected value when he does not

make any payments to the seller and the seller keeps the good. Note that theinterpretation of outside options is not standard in our model. We could alsosetU0s(s) = s andU0b(b) = 0. This would imply that preferences are notinterdependent when there is no trade.

The IIR requires an agents net utility given incentive taxes to be non-negative for all of that agents types:

Ui(, i) U0i(i) = Ui(, i) +

ii

Qi(s, )ds U0i(i) 0

for all i M, i i.

This is only true if

Ui(, i) + mini

ii

Qi(s)ds U0i(i)

0.

It is easy to see that the IIR constraint is binding for the lowest possibletype of the buyer and for the highest possible type of the seller. Note that

for the buyer individual rationality is satisfied if and only if Ub

(, b

) s

12

sdFs(s). For the seller, individual rationality requiresUs(, s) ss

Qs(a, )da 1+2 s

0. The expected surplus using our formulation inthe paper can be written as

B()

i{b,s}

xX

yx()

Vi(x, , )

Vi(x, , )

i

1 Fi(i)

fi(i)

dF()+

Ub(, b) Us(, s).

Repeating the same arguments, an IE mechanism maximizes

maxy

i{b,s}

xX

yx()(Vi(x, , ) +Vi(x, , )

iFi(i) 1

fi(i) +

0i

1 i(i)

fi(i) +

1

0i

Ii(i)

fi(i)

dF() (17)

0 (, ) (18)

0 , (19)

0 = ( )(, ), (20)


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where

Ii(i) =

1 ifi


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Table 1: Relationship between the ex ante efficient trade level and the degree of

social concerns in preferences

b s

0 1 00.3 1.15 0.08

0.6 1.3 0.17

1 1.45 0.25

the buyer adjusts her total valuation downward. They are willing not to tradeeven if trade is beneficial to both parties to get more favorable total payoffs.This may not be the case in IE mechanisms depending on welfare weights.

Table 1 summarizes the relationship among the resource feasibility La-grangian multiplier (), the degree of altruism , and information state ofthe economy = (b, s) for which trade occurs.

Note that the probability ofex anteefficient trade is equal to the prob-ability of efficient trade when = 0. For this case, agents only care aboutthe total valuations but do not care about the transfers, so the problem isequivalent to finding efficient mechanisms. This will not be true for all IEmechanisms since welfare weights might be type dependent. We can con-clude from the table that the probability of trade decreases as increases

since there will be less (s

, b

) for which trade occurs as increases. This im-plies altruistic agents trade more often than selfish agents. Moreover, selfishagents are more willing to risk losing beneficial trades to get a more favorablepayment than unselfish agents. The following result states that this observa-tion can easily be extended to all IE mechanisms.

PROPOSITION 5: Trade occurs more often as the degree of altruism in preferencesgoes up (e

for all1 > > >0).

The above result implies that there are more information states of theeconomy ( ) where it is IE to trade as decreases. Note that agents donot trade when it is not optimal to trade (b s . Moreover, the probability ofefficient trade is equal to one if full social preferences are in action ( = 0).This is the most important difference of our result with the Myerson and

Satterthwaite (1983) impossibility result with selfish agents.

5. Concluding Remarks

In this paper, we have characterized IE mechanisms for Bayesian envi-ronments with interdependent preferences. We mostly concentrated on


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regular problems where we assumed that monotonicity constraint is notbinding and provided a sufficient condition for regular problems. The ex-tension of characterization to irregular problems remains open. We also pro-

vided applications that show the properties of these mechanisms for bothpublic and private goods environments.Our initial intention was to extend our analysis to the case where the

individuals share prior claims to the objects (dissolving a partnership). Inthat case individual rationality constraints are type specific and the determi-nation of buyers and sellers is endogenous. Moreover, individual rationalityconstraints bind in the interior and this interior point (or region) is alsoendogenous. This creates difficulties in separating virtual valuations fromthe allocation rule, and hence virtual valuations are also endogenous. Our

conjecture is that the set of initial shares for which efficient dissolution of apartnership is possible extends as the degree of altruism in preferences goesup. Extending the formulation to this problem is an open question. We didnot have this problem in our formulation because individual rationality con-straints are binding for the lowest or highest types of agents for all incentivecompatible mechanisms.

One possibility for future research is to consider a model where thesocial concerns of agents are also private information. Then, types aremultidimensional. This extension appears to be a difficult open question

since the problems with multidimensional analysis are well known in themechanism design literature. Another possibility for future research is toconsider a mechanism design problem with Fehr and Schmidt (1999)type of preferences where individuals are inequity averse. This complicatesthe mechanism design problem since this type of preferences introducesdiscontinuities.

Appendix

The following Lemmas are used for the proof of Theorem 1.

LEMMA 1: A mechanism is IIC if and only if

(si i) (Qi(si, ) Qi(i, )) 0 for all si,i i, (A1)

Ui(, i) = Ui(, i) +

ii

Qi(s, )ds, (A2)

where

Qi(i, )

i

xX

yx()Vi(x, , )

i dFi(i).


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Proof: () Letsi > i. IIC impliesUi(, i) Ui(, i, si) andUi(, si) Ui(, si, i), where

Ui(, i, si) =Ui(, si)ixX

yx(si, i)Vi(x(si, i), si, i, )dFi(i)

+

i

xX

yx(si, i)Vi(x(si, i), , )dFi(i)

and

Ui(, si, i) = Ui(, i) i xXyx()Vi(x(), , )dFi(i)+

i

xX

yx()Vi(x(), si, i, )dFi(i).

This implies Qi(si, ) Ui(,si)Ui(,i)

sii Qi(i, ) and hence Qi(i, )

is nondecreasing. Letting si i implies Ui(,i)i

= Qi(i, ). Then

Ui(, i) = Ui(, i) +

i

i Qi(s, )ds.

() Now suppose 1 and 2 hold. Then Ui(, si) Ui(, i) =sii

Qi(s, )ds (si i)Qi(i, ). This implies, repeating the constructionbackwardly in the necessary part, Ui(, i) Ui(, i, si) and Ui(, si) Ui(, si, i). See also Rochet (1987) for a similar result for linear environ-ments with selfish agents.

LEMMA 2: An IIC mechanismis IIR if and only if for all i N,Ui(, i) 0.

Proof: IIR is satisfied if and only ifUi(, i) 0 for alli, i. By IIC

Ui(, i) = Ui(, i) +

ii

Qi(s, )ds 0 for alli N, i i.

That is, it requires

minii

Ui(, i) +

i

iQi(s, )ds

0

Ui(, i) + minii

ii

Qi(s, )ds

0 Ui(, i) 0 for alli N.

The other direction is trivial sinceVi(x, i) is monotone increasing in i.


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LEMMA 3:

i

i

i(i)Ui(, i) +

i

i

Qi(s, )ds dFi (i)

= 0i

Ui(, i) +

ii

1 i(i)

fi(i)

Qi(i, )dFi (i)

.

Proof: By changing the order of integration we get:

LHS = 0iUi(, i) +

i

iQi(s, )

i

s

i(i)dFi (i)

ds

= 0i

Ui(, i) +

ii

Qi(s, )(1 i(s))ds

= 0i

Ui(, i) +

ii

1 i(i)

fi(i)

Qi(i, )dFi (i)

.

Proof of Theorem 1: Directly follows from Lemmas 1, 2, and 3. (2) is EABB,

(3) is IIR, and (4) is the first part of IIC.

LEMMA 4:

Ni=1

0iUi(, i) = N

i=1

Ui(, i)

= (, ).

Proof: Let = maxiN{0i}. Define also K = {k| = 0k, k N}, the setof agents who have the highest ex ante welfare weight, and M = {m| >0m, m N}, the set of agents whose welfare weights are lower than thehighest ex ante welfare weight, where N = K M. There are two possiblecases:

Case 1: > . This implies for all i N, i >0 Ui(, i) = 0 IIRconstraints are binding for all agents lowest types.

Case 2: = . This implies for each k K, = = 0k k = 0

Uk(, k) 0 and for each m M, = > 0m m >0 Um(, m) = 0 IIR constraints are binding for all agents lowesttypes inMand the constraints are not binding for all agents inK.

From Cases 1 and 2, if Ui(, i) = 0 for some i M N then forall i M ex ante welfare weights are equal to = = 0i. This implies


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Ni=1

0iUi(, i) = N

i=1Ui(, i) = (, ). The second equality fol-

lows by EABB constraint which is always binding.

Proof of Theorem 2: It follows from Lemma 4 and the discussion for Cases1 and 2 as stated above. Suppose the type-dependent welfare weights aresuch that0i = > 0j for all j N\{i} and . The payment function(if = 0) is given by:6

j =i, aj(j) =

j

xXy

x()V j(x, , )dFj(j)j

j Q j(s, )ds

,

(A3)

ai(i)=

i

xX y

x()Vi(x, , )dFi(i)(, )i

i Qi(s, )ds

.

(A4)

We know that IIR constraint is binding for the lowest types of all i. Thisimplies

lNU

l(, l) = Ui(, i) = (, ) since EABB is always binding.Then, ai(i) =

i

xX y

x()Vi(x, , )dFi(i) Ui(, i). By usingthe envelope condition, we get the payment function of the agent with

maximal welfare weight. This implies agent i is the residual claimant.For other possible welfare weights, we will need additional constants tofind the payment rule. Suppose we are in Case 1. This implies, (, ) =0 from EABB and

i

xXy

x(i, i)Vi(x, i, i, )dFi(i) = ai(i)from IIR. Therefore, we can uniquely solve for the set of expected paymentsof all agents minimum types. Suppose now we are in Case 2. The argumentfor each i Mis similar to Case 1. On the other hand, for each i K theIIR constraint may not be binding. Therefore, we need |K| constants to solvefor the payment function. Note that the agents with the maximal expected

welfare weight share the remaining surplus (or cost) to make the EABBconstraint binding. This implies agents in set K will be residualclaimants.

Proof of Theorem 3: The solution is constructed such that all constraintsother than monotonicity are satisfied. We only need to show that the solu-tion satisfies the monotonicity constraint. Suppose i si, x= argmaxmX

Ni=1W

i(m, , , i) and y = argmaxmX

Ni=1W

i(m, si, i, , i). This

6 If = 0, any appropriate incentive taxes that add up to zero will work since in thiscase agents only care about the average payment and we know that budget is alwaysbalanced.


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impliesx yby Assumption 1(a). By Assumption 1(b),

Qi(i, ) = i

xX

yx(i, i) Vi(x, , )i

dFi(i)

=

i

Vi(x, , )

i dFi(i) Qi(si, )

=

i

Vi(y, si, i, )

si dFi(i).

This impliesQi(i, ) is monotone increasing. Note that ifx= y,Qi(i, )is obviously monotone increasing since we initially assumed that the valua-tion functions are monotone increasing in type for all agents. Therefore, thesolution (9) satisfies all constraints in Theorem 1.

Proof of Proposition 3: Suppose the welfare weights are decreasing in

type. This implies

Ni=1

Fi(i)i(i)fi(i)

0) = Prob(|Ni=1 i + ( + 1N )Ni=1

Fi(i)i(i)fi(i)

K). Now consider > and whereyx=1(, ) = 1. It is

easy to see that Prob(yx=1 >0) Prob(yx=1 >0) since there is such thatN

i=1 i + ( + 1

N )N

i=1Fi(i)i(i)

fi(i) K

Ni=1

i + ( + 1N

)N

i=1

Fi(i)i(i)fi(i)

. Note also that if yx=1(, ) = 0 then yx=1(, ) = 0. Next con-

sider > and where yx=1(, ) = 0. Then Prob(yx=1 >0) Prob

(yx=1 >0) since there issuch thatNi=1

i + ( + 1N

)

Ni=1

Fi(i)i(i)fi(i)

K Ni=1 i + ( + 1N )Ni=1 Fi(i)i(i)fi(i) . Note also that if yx=1(, ) = 1then yx=1(, ) = 1. The proof for the case of increasing welfare weights isalso similar.

Proof of Proposition 4: Given welfare weights and priors, let ={|

jW j( , , i) K} be the set of types where public good is produced

by an IE mechanism , given (, ). Let

= {|

jW j(, , i) K} be

the set of types where public good is produced bygiven (, ). Note that

for all

[0,

1] and all

, Ni=1 i K since efficiency, interim in-centive compatibility, and interim individual rationality are incompatible.This implies the adjustment term in modified virtual valuations is always neg-ative. Suppose without loss of generality < . We want to show that thereare more information states of the economy where the public good is pro-duced as the degree of altruism in preferences goes up,

. Suppose


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on the contrary there is such that and

. Then

N

i=1 i + +

1

N N

i=1Fi(i) 1

fi

(i

)

+0i

1 i(i)

fi

(i

) Kand

Ni=1

i +

+

1

N

Ni=1

Fi(i) 1

fi(i) +

0i

1 i(i)

fi(i)

. We also know that from first-order con-ditions. This implies

(, )

K dF() (, )

K dF() = 0,

where

(, ) =

Ni=1

i +

+

1

N

Ni=1

Fi(i) 1

fi(i)

dF().

Let, without loss of generality, = {|

jj >A(, K)} and

=

{|jj >B(, K)}. Since ( + 1

N )

Ni=1

Fi(i)1fi(i)

>( + 1N

)Ni=1

Fi(i)1fi(i) for all ,B(, K)>A(, K). This implies if

then ,contradicting our initial assumption.

Proof of Proposition 5: Given welfare weights and priors, let be the setof types where trade occurs given (, ) and

be the set of types wheretrade occurs given (, ). Let, without loss of generality, < (altruism inthe preferences increases). We want to show that

. Suppose on thecontrary there exists such that but

. (We know that either

or

). This implies

b s +1 +

2

Fb(b) 1

fb(b) +

0b

1 b(b)

fb(b)

Fs(s)

fs(s)+

0s

s(s)

fs(s)

0

and

b s +1 +

2

Fb(b) 1

fb(b) +

0b

1 b(b)

fb(b)

Fs(s)

fs(s)+

0s

s(s)

fs(s)

0.

This is only possible if > . This implies (, ) = 0. Since >

and trade does not occur in , we have

b s +1 +

2

Fb(b) 1

fb(b)

Fs(s)

fs(s)

b s +

1 +

2

Fb(b) 1

fb(b)

Fs(s)

fs(s)

0.


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We know that trade occurs for in . Hence,

(, ) =

b s +

1 +

2 Fb(b) 1

fb(b)

Fs(s)

fs(s)

dF() (, ) = 0.

This is a contradiction since is chosen by the algorithm such that(, ) 0.

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