The Principal-Agent Relationship with an Informed Principal: The Case of Private Values Author(s): Eric Maskin and Jean Tirole Source: Econometrica, Vol. 58, No. 2 (Mar., 1990), pp. 379-409 Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/2938208 . Accessed: 13/02/2011 12:40 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=econosoc. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Econometrica,Vol. 58, No. 2 (March, 1990), 379-409

THE PRINCIPAL-AGENT RELATIONSHIP WITH AN INFORMED PRINCIPAL: THE CASE OF PRIVATE VALUES1 BY ERIC MASKIN AND JEAN TIROLE We analyze the principal-agent relationship when the principal has private information as a three-stage noncooperative game: contract proposal, acceptance/refusal, and contract execution. We assume that the information does not directly affect the agent's payoff (private values). Equilibrium exists and is generically locally unique. Moreover, it is Pareto optimal for the different types of principal. The principal generically does strictly better than when the agent knows her information. Equilibrium allocations are the Walrasian equilibria of an "economy" where the traders are different types of principal and "exchange" the slack on the agent's individual rationality and incentive compatibility constraints. KEYwoRDs:Contract, principal-agent relationship, information revelation, general equilibrium, sequential games of incomplete information.

1. INTRODUCTION

screening (also called the theory of adverse selection or discrimination) represents a major accomplishment of the economics of information in the last two decades. This theory is often cast in a framework with two parties, a principal and an agent. The principal offers a contract, which the agent decides to accept or reject. The agent has private information about some parameter of his utility function. This parameter determines his "type." The parameter affects the principal's payoff at least indirectly, since the agent's type establishes the class of contracts that he will accept. The literature has developed this model both in the abstract (see Laffont-Maskin (1982) and Guesnerie-Laffont (1984) for unified treatments) and as applied to a variety of interesting economic problems, e.g., labor contracts, optimal taxation, price and quality discrimination, insurance contracts, educational screening, auctions, public goods, and regulation of monopoly. An important hypothesis of the usual model is that the principal is "uninformed," i.e., does not possess private information when contracting. Thus, the asymmetry of information is one-sided. One can think of many circumstances, however, where such an assumption is too restrictive. For example, in the literature on public good mechanisms (see Green-Laffont (1979) for a comprehensive bibliography) the informational deficiency usually emphasized is the government's (principal's) lack of knowledge of consumers' (agents') preferences. But at the time the government institutes a mechanism for eliciting those preferences, it may well know more than consumers about the cost of supplying

THE DEVELOPMENT OF THE THEORY of

1 This research was supported by the Sloan Foundation, the U.K. Social Science Research Council, and the National Science Foundation. We are grateful to Drew Fudenberg, David Kreps, Roger Myerson, and a referee for helpful comments. We are much indebted to Andreu Mas-Colell for considerable technical advice, and to a second referee for several important insights. This paper was formerly entitled "Principals with Private Information, I: Independent Values." 379

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the good. Alternatively, when a principal such as the Department of Defense deals with contractors (agents) to develop a missile, it may well have special knowledge of the weapon's strategic value. Similarly, a regulator may have private information about the demand for the regulated good when devising a regulatory scheme for a firm it controls. As for private sector examples, a monopolist might have exclusive information about the quality of the good it sells when offering a warranty/maintenance contract to its customers. Similarly, a manufacturer that proposes a franchising agreement to a new retailer could well have superior data about the state of demand. In keeping with the bulk of the literature, we restrict attention in this paper and its sequel to the case where only one party has a hand in designing the contract. Following standard terminology, we designate the "principal" as the contract "designer" (proposer) and refer to the party that accepts or refuses the contract as the "agent." We deviate from convention, however, by assuming that the principal, as well as the agent, has private information. This assumption complicates contracting because by her very proposal, the principal may reveal some of what she knows. The revelation of information by contract proposal was emphasized by Myerson (1983) in his seminal article (see also Crawford (1985)). Myerson and Crawford, however, studied the principal-agent relationship using techniques drawn from cooperative game theory. In particular, Myerson was especially concerned with establishing the nonemptiness of the core. The purpose of our project, by contrast, is to develop a noncooperative theory of the principal-agent relationship when the principal has private information. Section 2 lays out the "principal-agent" or "contract proposal" game, which comprises three stages. Two parties meet after having learned their private information (type). In the first stage, one party, the principal, proposes a contract. The contract is itself a game form in which each party is given a finite set of messages from which to pick and that specifies an action (e.g., producing some level of output) to be taken by the agent and a transfer from the principal to the agent for each pair of messages chosen by the two parties. The agent accepts or refuses the contract in the second stage. If he refuses, the game is over. If he accepts, players proceed to the third stage, where they carry out the contract, i.e., choose their messages and implement the corresponding action and transfer. We assume that actions and transfers are observable (and verifiable), thereby ruling out any moral hazard. Notice that our framework is the same as the classic screening model,2 except that the principal has private information 2

By the "classic screening" model, we mean a model of asymmetric information where the "contract" is either chosen by a player without private information (as, for example, in optimal income taxation (c.f., Mirrlees (1971)), monopolistic nonlinear pricing (c.f., Mussa-Rosen (1978), and Maskin-Riley (1984)), and optimal regulation (c.f., Baron-Myerson (1982) and Laffont-Tirole (1986)), or else is negotiated before the asymmetries arise (c.f., Grossman-Hart (1981)). The standard "signaling" model, by contrast, entails an informed party proposing the contract and an uninformed party accepting or rejecting it (c.f., Spence (1974)).

PRINCIPAL-AGENT RELATIONSHIP

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when contracting.3 This information, however, is important since it will typically affect the ultimate outcome. In this project we distinguish between the cases where the agent cares and those where he does not care "directly" about the principal's type or information. The former case is that of common values. In the latter case, where we say that values are private, the agent's expected payoff is a function only of the principal's behavior, not of her information. Formally speaking, privateness means that, holding the principal's behavior fixed, her information parameter is an argument neither of the agent's von Neumann-Morgenstern utility function nor of the probabilities he assigns to the variables entering his utility function.4 Of course, even in this case, the agent typically cares indirectly about the principal's information because the outcome of the third stage (i.e., the determination of an action and transfer) may depend on the principal's message, which in turn is influenced by her information. In this paper we shall deal exclusively with private values. This hypothesis seems a good approximation for the public good, procurement, and regulation examples mentioned above. For instance, the weapons contractor cares only about its profit and not per se about the defense value of the missiles it creates. The reader may wish to keep these three examples in mind as paradigms of the sort of situation we are trying to model. By contrast, the common-values case, where the agent's payoff depends directly on the principal's type, is illustrated by our monopoly and franchising examples. Specifically, the consumer of a particular good is ordinarily concerned directly about the quality of that good. We take up this case in Maskin-Tirole (1988). Section 3 demonstrates that the equilibrium contract in our model generally differs considerably from that of the standard principal-agent framework (where only the agent has private information). Indeed, the principal profits from the agent's incomplete information about her type. To see why this is so, note that when the principal proposes a contract, she does so subject to two kinds of constraints. There is the requirement that the contract should not leave the agent worse off than with no contract, i.e., the individual rationality (IR) constraint. There are also constraints ensuring that, when the contract is carried out, the agent behaves in the appropriate way given his private information. These are the incentive compatibility (IC) constraints. Now, when the agent knows the value of 3In this respect,our frameworkis a synthesisof the screeningand signalingmodels.In this paper, however,we concentrateon the case of "privatevalues,"whereasthe signalingliteratureto date has concentratedprimarilyon commonvalues(see the next severalparagraphsfor the distinction). 4To see that, in general,we have to ruleout the parameter'saffectingthe agent'sprobabilities(as well as enteringhis utility function),thinkof a model of a moralhazardin which output depends stochasticallyon the agent's(unobservable)effortand wherethe principalhas privateinformation the agent'sutilityfunctiondoes not dependon the about this stochasticrelationship.Conventionally, principal'sinformation,but ratheron his effortand monetaryreward.Nonetheless,the principal's information does directly affect the agent's probabilisticbeliefs about output and, hence, his monetarytransfer.Thusthe agent'sexpectedpayoffis, afterall, a functionof the principal'stype.We concludethat such a modelis an instanceof commonvalues.

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the principal'sinformationparameter(the case of "full information"),the IR and IC constraintsmusthold individuallyfor each type of principal.Withincomplete information,however,they need hold only in expectationover the principal's types. Thus, a given type of principalcan raiseher utility above the full-information level (where all the constraintsmust be satisfied)by violating some constraints,as long as these violationsare offset by the other types. In fact, we can think of differenttypes of principalas trading"slack"with one another:one type, say, accepts some slackon the IR constraintin exchangefor being allowed to violate an IC constraint,whereasanothertype does just the opposite.As we show in Section 3A, generically(in the space of utility functions)there exists a contractin which all types of principaldo strictlybetterthan in the case of full information(Proposition1). This result dependscruciallyon the private-valuesassumption.Consider,by contrast,a Spencianlabormarket(c.f., Spence(1974))in whichthe "principal"is an employeeof eitherhighor low productivity.In thiscase the agent's(employer's) payoff certainly depends directlyon the principal'stype. It is clear, moreover, that the high productivityemployee is likely to be hurt by the employer's incomplete information: either she will find herself "pooled" with her low productivitycounterpart(in which case her wage will fall short of her marginal product) or else she will have to undertakecostly signalingactivity(e.g., education) to distinguishherself.5Thus in a common-valuesmodel, unlike one with privatevalues, thereis a conflictamongthe differenttypes of principal. We can say much more about the equilibriumof our three-stagegame than merelythat the differenttypes of principaldo betterthan underfull information. Indeed, to continuethe tradinganalogyintroducedabove, considerthe fictitious pure-exchangeeconomyin which the tradersare the differenttypes of principal and the goods exchangedare the slack variables.A trader'sinitial endowment consists of the values of these slack variablesunderfull information(i.e., zero). For reasons exactly parallelingthe usual competitive analysis, a Walrasian equilibriumalwaysexists for this economy(Proposition2). Strikingly,moreover, the Walrasianallocationsare precisely the perfectBayesianequilibriumoutcomes of our three-stagegame(Propositions6 and 9). we can readilyestablishthe genericlocal From this Walrasiancharacterization, uniqueness (Proposition 10) and Pareto optimality (Pareto optimality is, of course, constrainedby the fact that the agent also has privateinformation)of equilibrium.Indeed, a strongconcept of Paretooptimalityoffers an alternative characterizationof equilibrium.For a contractto be feasibleit must satisfy the IR and IC constraintsin expectation.The "expectation,"of course,dependson the agent's beliefs about the principal.A feasible contract is strongly Pareto optimal (from the point of view of the differenttypes of principal)for given 5Such an outcome is impossible with private values. A given type of principal can always simply propose the contract that would obtain if the agent knew her type (the "full information" contract), and the agent will accept regardless of his beliefs. In the labor-market example, by contrast, the employer would reject the full-information contract proposal of the high-productivity employee if he thought there was a chance the employee had low productivity.

PRINCIPAL-AGENT RELATIONSHIP

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beliefs if there exists no other feasible contract, evenfor differentbeliefs, that Pareto dominatesit. A Walrasianallocationis stronglyParetooptimal(Proposition 3, our analog of the First FundamentalTheoremof WelfareEconomics). Moreover,every strongParetooptimumis Wairasian(Proposition4, the Second Welfare Theorem). Hence, given the above-mentionedequivalence between Walrasianequilibriaand the PBE's of our game, the same equivalenceholds between the strongParetooptimaand the PBE's(Proposition7).6 To reap the gain from the agent's incompleteinformation,when values are private, the principal must refrain from revealing her type at the contract proposal stage (otherwise,the IR and IC constraintsmust hold for that type, rather than just in expectation).To accomplishthis concealment,the various types of principalhave to "pool,"i.e., proposethe same contractin equilibrium.7 After our main analysis,we considerin Section4 the special,but often-studied case wherethe principaland agenthave quasi-linearobjectivefunctions(utilities that are additivelyseparableand linearin transfers).In this nongenericcase, the Walrasian equilibriumof the fictitious economy involves no trade and the (unique)equilibriumoutcomeof our contractproposalgame coincideswith that of the standardprincipal-agentmodel. In other words,with quasi-linearpreferences, the principalneithergains nor loses if her informationis revealedto the agent before contracting.Section5 concludes. 2. THE MODEL

We now describethe model. In the conclusion,we argue that severalof our simplifyingassumptionscan be relaxedwithoutaffectingthe results. A. ObjectiveFunctionsand Information

There are two parties, a principal and an agent. The principalhas a von utility functionV(y, t, a), wherey is an observable(and Neumann-Morgenstern verifiable8)action, t is a monetarytransfer(whichcan assumenegativeas well as positive values)fromthe principalto the agent,and a is a parameterrepresenting the principal'sprivateinformationor "type."We shall supposethat y, t, and a 6 This result togetherwith Proposition10 suggestshow powerfulthe concept of strong Pareto optimalityis: there is a continuumof ordinaryParetooptimabut, generically,only finitelymany to the priorbeliefs. strongParetooptimacorresponding 7 The mereobservationthat thereexistsa poolingequilibrium of the contractproposalgameis, by itself, a triviality and holdsirrespectiveof whethervaluesare privateor common.Indeed,it is just a reflection of the "InscrutabilityPrinciple"of Myerson (1983), which notes that any possible equilibriumoutcome arises from some pooling equilibriumif the set of available contractsis sufficientlylarge.The realsubstanceof our poolingresultis thatsuch a separatingequilibriumis not possible when values are privateand the principalalso has privateinformation.Generically,in this latter case, all equilibriaentail some pooling. In particular,no subset of types of principalis completelyseparated.(If some subset were completelyseparated,its memberswouldnot tradeat all with the complementarysubsetin Walrasianequilibrium,whichis genericallyimpossible.) 8By " verifiable"we meanthatthe actionis observableby a thirdparty;thus,it can be specifiedby an enforceablecontract.

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are real numbers.In the case wherethe principalis a buyerof some good and the agent is a seller, one can think of y as the quantityof good deliveredto the principal.Nothing turns,however,on whetherthe action is, in fact, taken by the agent or the principal,since it is observedby both and can be specifiedby a contract.The functionV increaseswith y and decreaseswith t. It is continuously differentiableand concavein the pair (y, t) and strictlyconcavein y. The agent has a von Neumann-Morgenstern utility functionU(y, t, 0), where the informationparameter0 (a scalar) is the agent's type. That U does not depend on a embodies the assumptionof privatevalues and is an important assumption.(In contrast,our resultswouldbe unaffectedif V dependedon 0. See the conclusion.) U decreaseswith y and increaseswith t; it is continuously differentiableand concavein (y, t) and strictlyconcavein y. We will also assume that it decreaseswith 0: if 01 < 02, thenU(y, t, 01) > U(y, t, 02) for all (y, t). We shall suppose that in the absenceof a contractwith the principal,a "null" contracttakeseffectin whichthe agentobtainsreservationutility iu9Throughout the paper, superscripts(indexed by i) and feminine pronouns refer to the principal,whereassubscripts(indexedby j) and masculinepronounsapply to the agent. To guaranteethe existenceof equilibrium,we assumethat the feasibleactions and transferslie in compactand convexsets. Let ,udenote a probabilitymeasure on these sets. If, for example,I is discrete,,u{ y, t }) representsthe probability of action y and transfert. We will allow contractsto specify a measure, as an outcome.We thus permitrandomoutcomes. We assume that the parametersa and 0 are drawnfrom known and statistically independentdistributions.Parametera is knownonly to the principal,and O only to the agent. We suppose that each parametercan assume only finitely many values: a = al,..., a n with positive probabilities 7i, ..., 9"n such that Ei= 7Ti= 1, and 0 = 01 and 02 with positive probabilities Pi and P2 (Pl +P2 = 1). The restriction of the agent's parameterto two values is not essential. It ensures that only a single incentive compatibilityconstraint is binding (see Lemma1), whichis notationallyand expositionallyconvenient.As the readercan readily check, however,all our resultsrequireonly that at least two (IR or IC) constraintsbe binding. To simplify the notation,we define

Vi(.u) fV(y, t, a') d,u({y, t})

(i = 1 .. ., n)

and Uj(u)-= fU(y, t,90) d4({yy t)

(1 = 1,2).

9Thus, we can assume that the principal and agent always sign a contract, since the absence of a contract is just a special case of having one. Formally, we must assume that the null (y, t) pair (ordinarily y = t = 0) belongs to the feasible set.

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PRINCIPAL-AGENT RELATIONSHIP

Let t be the smallest transferfrom the principalto the agent (it may well be negative).We will assumethat given this transferthe agent is necessarilyworse off than without a contract,regardlessof his type or the action y: ASSUMPTION 1:

Uj(y, t, Oj)< u for all y and j = 1, 2.

We will suppose also that, regardlessof the values of 0 and a, there exists a feasible action and transferthat both partiespreferto the null contract,i.e., to the absenceof a contract. B. The Principal-AgentGame

Let us describeour three-stagegame in detail. In the first stage the principal proposes a contractor mechanismin the feasibleset M (we will use the words "contract"or "mechanism"interchangeably).A mechanismm in M specifies (i) a set of possible messagesthat each partycan choose and (ii) for each pair of messages sP and Sa chosen simultaneouslyby the principaland agent, respectively, a correspondingmeasureji on the set of deterministicallocations(y, t).10 Thus, a mechanismis a game form that selects a (random)outcomeconditional on a pair of (payoff-irrelevant) Observethat,becausethe principal,as messages.11 well as the agent, can make announcements,she may be able to revealinformation at the third stage (see below) as well as at the contractproposalstage. For the moment we let M denote the set of finite mechanisms(mechanisms where the numberof availablemessagesfor each party is finite) for simplicity. For technicalreasons,we will slightlyexpandthe set of allowablemechanismsin Section 3D.12 Notice that the set M includes the set of directrevelationmechanisms,in which both parties simultaneouslyannouncetheir types (not necessarilytruthfully). Hence, in a direct revelation mechanism (DRM), (sP,

sa) = (a,

0), where a

hat denotes an announcedvalue.We will makeconsiderableuse of these DRM's by repeatedlyinvokingthe revelationprinciplefor Bayesiangames(see DasguptaHammond-Maskin(1979) or Myerson (1979)). In the present context, this principleasserts that, for any mechanismand for given beliefs at the time that mechanismis about to be played (i.e., after it has alreadybeen accepted),any equilibriumof the mechanismcorrespondsto an equilibriumof some DRM in which announcementsare truthful. Observethat if a assumedonly one value(i.e., therewereno uncertaintyabout the principal), the revelationprinciple would imply that the principal could 10Becausethe outcomecontingenton sP and Sa can be random,the mechanismcan incorporatea correlatingdevice a la Aumann(1974). 11For simplicity,we are restrictingattentionto mechanismswhere there is a single round of messageschosen simultaneously.With no changein the arguments,however,we could extend our results,using the revelationprinciple,to mechanismswith morethanone round. 12 Specifically,we allowa thirdparty,as well as the principaland agent,to send messagesin these mechanisms.

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restrictattention to mechanismswhereshe sends no messagein the third stage. Becausethe conventionalassumptionin the literatureis that the principalhas no private information,we shall call a mechanismstandardif it consists simply of the agent'sannouncinghis type (so that principalannouncesnothing). In the second stage, the agent accepts or refuses the contractofferedby the principal.He obtainshis reservationutilityif he refuses.'3If he accepts,the two parties play the proposed mechanismin the third stage (for instance, they announcetheir types if the mechanismis direct),and the allocationcorresponding to their thirdperiodmovesis implemented. The principal's strategy in the three-stagegame consists of a choice of mechanismand a choice of message(sP) for each mechanismin M. The agent's strategyconsists of (i) the decisionto accept or rejectthe mechanismand (ii) a choice of announcement(sa) in the mechanism.Both the agent's decisions are contingenton the mechanismproposed. We are interestedin the perfectBayesianequilibriaof the overallgame.14 In our framework,such an equilibriumis a vector of strategies-one for each type of player(in our model,thereare n + 2 types)-and a vectorof beliefs about the other player's type15 at each informationset in the game tree such that (i) the strategies are optimal (i.e., at all points in the tree each type is maximizing expected utility given beliefs and the other types' strategies);(ii) beliefs are derivedfrom Bayes'rule given observedbehaviorand the equilibriumstrategies; and (iii) the principal'sbeliefsabout the agentat the end of the firststageremain the prior beliefs (regardlessof her proposal)and the agent's beliefs about the principalare the same at the end of the second stage as at the end of the first.16 Thus, in particular,we assume that the agent updates his beliefs about the principal's type using Bayes' rule, after observingthe contract she proposes. Similarly,we suppose that the principalrevises her beliefs appropriatelyafter observingthat the agent has acceptedthe contract.In the continuationgame of the thirdstage, theremay, of course,be multipleequilibria.We supposethat the 13

In this respect, our model differs from that of the multiperiodbargainingliterature(e.g., Admati-Perry(1986),Fudenberg-Tirole(1983),and Sobel-Takahashi(1983),whereinit is typically assumedthat the seller(principal)cannotpreventherselffrommakinganotherofferif the buyerturns her downinitially.Note also thatthereis no conflictbetweenour "pooling"resultand the separating equilibriaof the noncooperative bargainingliterature.A poolingequilibriumdoes not implythat the differenttypesof principalend up with the sameallocation;we can thinkof a contractas a schedule of allocations-one for each type.The differenttypesself-selectin the thirdstage. 14 Note that becausethe set of finitemechanisms is itself infinite,so is the strategyspace for the principal.Thus standardequilibriumexistencelemmas do not apply. Because the second stage continuationequilibriaare sequential(see footnote 16), however,standardresultsensurethat they exist and that theircorresponding payoffsareupperhemicontinuouswith respectto beliefs. 15The principal'sbeliefs are the probabilitiesthat she assigns to 6; the agent'sbeliefs are the probabilitiesthat he assignsto a. 16 Actually,with requirement (iii) (which,in effect,requiresthata player'sbeliefsaboutthe other's type are not affectedby his own actions),our definitionof perfectBayesianequilibriumis somewhat stronger than the usual definition.Indeed, as Fudenberg-Tirole(1989) show, conditions(i)-(iii) would imply that the equilibriumis sequentialif the principal'sstrategyspacewerefinite.

PRINCIPAL-AGENT RELATIONSHIP

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players can coordinateover these equilibriaby meansof some public randomizing device17 such as a coin flip. If the coin turns up heads, they play one equilibrium;if tails, they play another. Thus, in the third stage, we permit (publicly)correlatedequilibria. We denote by = {frTi}.' the agent'spriorbeliefs about the principal'stype and, by

VT=

{ 77 }7

his beliefs after the principal has proposed a contract. We

shall call these latter probabilitieshis posteriorbeliefs. To study the set of equilibriumoutcomes,we will use the revelationprinciple.For given posterior beliefs Ir=

{ r}7="

(such that Eil7T = 1), any outcome of the continuation game

between the principaland the agent is also the outcome of a direct revelation game in which, in equilibrium,both partiesannouncetheirinformationparameters truthfullyand simultaneously.In Sections3 and 4, we will firstconstructthe strategiesalong the equilibriumpath, and then consideroff-the-equilibrium-path proposals m for given posteriorbeliefs sz.Ratherthan study the equilibriumof the game described by m, we will instead work with the equivalent direct mechanism,u, whereIL is the (random)outcomeimplementedif the principal announcestype i and the agent announcestype j. C. The Case of Full Information

For reference,we firstexamineequilibriumwhen the principal'sinformationis common knowledge,i.e., the agent knows the value of a before contracting.We call this the full informationcase (the principal,of course, does not know the agent's type, but, since this feature is maintainedthroughoutthe paper, the terminologyshould not createconfusion).This is merelythe standardscreening set-up (see footnote 2). Let us assume that a = a'. From the revelationprinciple,we know that the equilibriumallocationcan be attainedby a standardDRM ,t'. where,in equilibrium, the agent revealshis type truthfully.The outcome ,ji (j = 1,2) that the contract specifies when the agent announcestype j must satisfy two types of constraints.First, the agent must be willing to accept the contract.That is, it must satisfy the individualrationality(IR) constraints:for j = 1,2, U)(,)> . u1 Second, the type j agent must tell the truth. This gives rise to the incentive compatibility(IC) constraints:for all j, k: Uj(O, ) > Uj(kk). Actually, in an optimalcontract(one maximizingthe principal'sutility),only two of these four constraintsare binding.Becausethe agent's utility decreases with his type, only the IR constraintfor an agentof type 2 is required;the other holds automatically.Moreover,this monotonicityof utilityimplies(see Lemma1 below) that only the type 1 IC constraint(U1(,4) > U(,u')) can be binding.Thus, when her informationis common knowledge,a prncipalof type i proposes a 17 The technicalreasonfor allowingpublicrandomization is

to ensurethat the equilibriumpayoff

set of the continuation game is convex. Note that this randomization is in addition to that already built into the mechanism (see footnote 10).

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ERIC MASKIN AND JEAN TIROLE

contract { pi, [/' } that solves the full informationprogram: 2

Max E pivi( ,uj) such that

(F')

{it', j=1

IRi: U2GLi ) ku ICi: Ul(pil)

pi

>, Ul(Pi2)

(7i),

where pi and y' denote the Lagrangemultipliersfor the IR and IC constraints, respectively.Becausethe sets of actionsand transfersare compactand the payoff functionsare continuous,a solutionto program(Fi) exists.We will denote it by { c,ii',pi,}. From Assumption1, it is clear that both constraintsare strongly binding (i.e., both p and ji are positive).Let v` E1jp1V(j'). We shall referto v' as the type i principal's full-information payoff, as to I=(jl1,..., jr!) as v=

(v31

vf)

as the full-information allocation and payoffs, respectively.

It is clear that, regardlessof the agent's informationabout the principal, v` providesa lower bound for the type i principal'spayoffin our three-stagegame. To see this, supposethatshe proposedthe mechanismiii'..Thenirrespectiveof his beliefs, the agent would accept the proposal because, by definition of the mechanism, he could guaranteehimself a payoff of at least iu by so doing. Moreover,againby definition,he will announcethe truth.Thus,by proposingthe mechanism 'ii, the principalensuresherselfthe payoff vi. 3. PERFECTBAYESIANAND WALRASIANEQUILIBRIA

Our goal is a completecharacterization of the equilibriaof the principal-agent game, but we begin with a simplerproblem:studyingthe contractthat would be proposed by a third party who maximizedan arbitraryweighted sum of the payoffs of the differenttypes of principal(Section 3A). We first show that, generically,this thirdpartycouldimplementa contractthat Paretodominatesthe full-informationallocation.It accomplishesthis by "pooling"the agent'sIR and IC constraintsover the differenttypes of principal,i.e., by havingthe constraints hold only in expectationratherthan for each singletype. This examinationleads naturally to a study of the Walrasianequilibriaand Pareto optima of the fictitious pure-exchangeeconomy where the tradersare the differenttypes of principaland "exchange"the slackvariablescorrespondingto the agent'sIR and IC constraints(Section3B). The relevanceof this competitiveanalysisis demonstrated when we establish that equilibriain the principal-agentgame exist (Section 3C) and correspondexactlyto the Walrasianallocationsof the fictitious economy. Equilibriathereforeinheritthe Paretooptimalityand local uniqueness propertiesof Walrasianallocations(Section3D). A. UnconstrainedPareto Optima

As the starting point of the analysis,let us consider the following thought experiment.Supposethat, ratherthan the principal,a third party proposesthe

PRINCIPAL-AGENT RELATIONSHIP

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contractbetween the principaland the agent.Assumethat it acts to maximizea weighted average of the payoffs of the differenttypes of principal,where the weights are nonnegativebut arbitraryand fixed beforehand.Suppose, furthermore, that after the contractis proposedand accepted,the principal'stype is made publicly known. The device of a third party is meant to rationalizethe objective function; if the principal were proposing the contract, she would certainlynot do so to maximizean arbitraryweightedaverage.The fact that the weights are fixed beforehandand do not depend on knowledgethe third party has avoids the complicatingpossibility that the proposal itself may reveal information.Note also that we do not take accountof any IR constraintson the part of the principal.18Finally,the assumptionthat the principal'stype becomes public knowledgeex post eliminatesthe issue of incentivecompatibilityfor the principalat the thirdstage.If a werenot publicknowledge,then any announcement the principalmadeat the thirdstagewould reflecther type; her announcement would have to satisfythe IC constraints.Underour hypothesis,by contrast, the contract can make the outcomedirectlycontingenton the principal'stype, without having the principalmakeannouncements. Given these assumptions,the thirdpartysolves the program: Max E wi(Ep1vi(Yi' )) suchthat (UPO)

IIR:

E

0lU2

( i2

T4

(p)

|IC: E owl(Ail) >,E 0iU1(11i2) (y). i

i

We call the solution to this programan "unconstrainedParetooptimum"(the term "unconstrained"refersto the fact that thereareno incentiveconstraintsfor the principal).Notice the agent'sbeliefs, 70, in program(UPO) may differfrom his prior 7r. The number w'

(Xwl

= 1) is a nonnegative weight for the type i

principal'spayoff. The constraintsare individualrationalityand incentivecompatibilityrequirements,given the agent'sbeliefs so. By omittingthe IC constraint for the type 2 agent,we are invokingthe familiarresult,provedin the Appendix, that this constraintholds automaticallyat an optimumof (UPO): LEMMA 1:

The IC constraintfor the type 2 agent is not bindingat an optimumof

(UPO).

Lemma 1 allows us to simplify the notation by reducing the number of constraints to two: one IR and one IC. One allocation that satisfies these constraintsautomaticallyis the full informationallocation,u. (since this alloca18 Recall that the type i principal can guarantee herself a payoff of i' by proposing ai.. Thus if type i were the proposer (or the third party were acting exclusively on her behalf), an equlibrium would necessarily have to satisfy the IR constraint of ensuring a minimum payoff of v'. In the framework of our thought experiment, however, no such constraint applies.

390

ERIC MASKIN AND JEAN TIROLE

tion satisfies IRWand IC' for each i). We next show, however, that generically, the full-information allocation is not unconstrained Pareto optimal. This proposition, proved in the Appendix, embodies much of the economic intuition for what follows. PROPOSITION 1: For a generic choice of utility functions (i.e., for an open and dense set, relative to the C' topology, of utilityfunctions satisfying the conditions of Section 2), there is, for any strictly positive posterior beliefs 7T (i.e., beliefs such that rI'> 0 for all i), an allocation that satisfies the IR and IC constraintsfor the agent with beliefs 7T and that Pareto-dominates the full-information allocation W (from the perspective of the different types of principal).

The idea behind the proof of Proposition 1 is readily summarized. The full information contract must satisfy the agent's IR and IC constraints for each type i of principal. If we introduce a small amount of slack - r' and - c' on these constraints (where r' and ci can be negative or positive), the type i principal can attain the payoff v=vi' + p5'r'+y'c',

where i' and yi are the shadow prices for the type i IR and IC constraints, respectively. Now, as long as

(1)

?ITiri < 0 i

and

c .icOi < O, i

the agent's constraints hold in expectation. Generically, we can choose { ri, c }i _ satisfying (1) such that v'- vi is nonnegative for all i and strictly positive for some i. The allocation, u,u: corresponding to this choice (i.e., that for all i solves program F' when the constraints are replaced by U2(,u'i)< iu- r' and Ul(,u() > c') Pareto-dominates ,. Indeed, we can think of ,u:* as being generUl(,uiated by the different types of principal "trading" slack variables. Under this interpretation, the full-information allocation corresponds to autarky. REMARK 1: For there to exist gains from trade, it is clear that at least two constraints for the agent must be binding. Hence, there must be at least two types of agent.

REMARK2: Proposition 1 accords with the observation in Hirshleifer (1971) that the premature disclosure of information may destroy advantageous trading opportunities. It goes well beyond Hirshleifer's analysis, however, by the demonstration that forestalling disclosure makes possible improvements that are Paretian for the different realizations of the principal's type (Hirshleifer considers only ex ante improvements, i.e., moves that improve welfare in expectation over the various types). As we will see, equilibria of the three-stage game turn out to be efficient in a sense much stronger than unconstrained Pareto optimality. A UPO allocation is

391

PRINCIPAL-AGENT RELATIONSHIP

AI

1' Locus of UPO payoff

I>

belief

s for

s 7r

Locus of UPO payoffs for beliefs ir

\ .l\

__B

12~~~~~~~~~~~~s

VI

I,V 2)

FIGURE1(a).-The generic case. V2

A

W*consists of the line segments connecting A and B to (vI ,v 2) B

(v,v

2)

VI

FIGURE1(b).- W* in the quasi-linear case.

defined for fixed beliefs,whereasa strong unconstrainedPareto optimum (SUPO) allows beliefs themselvesto be control variables.(See the Remark following Proposition5 for an indicationof how much strongerSUPO is than UPO.) An allocation ,u is SUPO for beliefs 7rif (i) it is UPO for those beliefs, and (ii) no other UPO allocation j: for any beliefs 'T Pareto-dominates,u, where the Pareto-domination must be strict (i.e., EjpjV'( i') > EjpjV i(,up) for all i) if 7ris

not strictlypositive.The set of SUPO payoffvectorsis thus given by

W* = {(v1,... vI)I there exists a UPO allocation ,: with beliefs ST such that vi = EjpjV (#u')for all i; moreover,there exist no other UPO allocation i, for beliefs q?,such that X1pV'(jVi',)> :V'(/,u)

for all i, with strict inequality for

some i and whereall inequalitiesare strictif s? is not strictly positive}. The set W* is thereforethe "outerenvelope"of the UPO payoffloci as beliefs vary. (See Figurela. Figurelb depictsthe nongenericquasi-linearcase studiedin Section 4.) Notice that the conceptof SUPO requiresan allocationto be strictly

392

ERIC MASKIN AND JEAN TIROLE

Pareto-dominated(i.e., each type of principaldoes strictlybetterin some alternative allocation) to fail as a candidatefor an optimumwhen the beliefs for the alternativeallocationare not strictlypositive.Thus,it is possiblefor one SUPO allocation to weakly Pareto-dominateanother (see Figure la, in which the portions of W* that lie along the axes exhibitParetodominance). B. The Fictitious CompetitiveEconomy

Let V,(r', ci) denote the type i principal'sindirectutility when there is slack -

r and

c' in the IR1 and IC' constraints, respectively. Thus Vj(r', c') is the

-

maximizedvalue of (F*i): (Max

pjVi(iji)

suchthat

(F*)2)lUlGel)

>, UG(tO2 - C .

Suppose that the type i principalis allowedto "buy"negativeslack (i.e., to sell slack) in the IR1and IC1constraintsat pricesp and y, respectively,subjectto the "budget"constraintthat the value of the negativeslack purchasedbe nonpositive. She then solves: (Di)

( Max Vji(ri,c')

subjectto

{r',c'}

k

pr' + yc' <

o.l9

Let Di(p, y) denote the type i principal's"demandcorrespondence,"i.e., the solution to the program(D'). We thus envision a competitive,pure exchange economy where the tradersare the differenttypes of principaland buy and sell slack. Althoughwe have not (yet) restrictedthe (ri, ci) pairs to a compactset, it is clear that a solution to (Di) exists, since the set of feasible pairs (y, t) is compact. In fact, a solution to (Di) must satisfy the budget constraintwith equality: LEMMA 2: If (?ri,c^i)eD1(p,y), then pr^?+yc =O.

The proof of Lemma2 is standard(see Maskin-Tirole(1986)). A Walrasianequilibriumof this fictitiouseconomyis a pair of positiveprices (p, y), and a choice of negativeslackvariables20(ri, ci) for each type i such that: ITiri = 0, (2) (3)

Tici=

o,

i

in program (D') already incorporates a O9Of course, the indirect utility function, VW(r%,c'), maximization over allocations ,u. 20 Note that, by referring to r' and c' as "negative slack" variables, we do not mean that their values are negative. Rather, we are saying only that - r' and - ci are slack variables.

PRINCIPAL-AGENT RELATIONSHIP

393

and (4)

(ri,

cl) E

DV(p, y).

Conditions(2) and (3) are "marketclearing"requirements,which ensurethat the "average"amountof negativeslack demandedfor each constraintequal the averagesupply, zero. Condition(4) simply requiresthat each trader'schoice of slack maximizeher (indirect)utilitygivenher budgetconstraint.We next observe that a Walrasianequilibriumexists in our model for reasonsanalogousto those in the classicalcompetitivemodel. PROPOSITION 2: There exists a Walrasian equilibriumof the fictitious economy relative to any beliefs qr.

The proof (which can be found in Maskin-Tirole(1986)) is standardfrom generalequilibriumtheory.It sufficesto checkthat the utilityfunctionsV/'(r', ci) satisfy the requisitecontinuityand concavitypropertiesand then to apply the usual Debreu (1959) techniques. Just as an ordinaryWalrasianequilibriumis Paretoefficient,so an equilibrium of our fictitiouseconomyhas attractiveefficiencyproperties. PROPOSITION 3: A Walrasianequilibriumof thefictitious competitiveeconomy is strongly unconstrainedPareto optimal (SUPO).21

Notice that a Walrasianallocation iKis SUPO even when the corresponding beliefs s' are degenerate,i.e., s' = 1 for some i. Now, with suchbeliefs,the type i principal's utility from this allocation is just the full informationlevel, vi. Therefore,becausegenericallythe full informationallocation ii: is not SUPO,we concludethat (generically)at least one othertype of principaldoes strictlybetter with ,u: than with ,-i.

Propositions2 and 3 togetherimply that there is a SUPO allocationfor any beliefs 7'. This result relies importantlyon the private-valuesassumption.By contrast, consider the (common-values)labor marketexampleof the introduction. In that model, it is readilycheckedthat thereexist "pessimistic"beliefs on the part of the agent (beliefsthat assigna comparativelyhigh probabilityto the "bad" type of principal)relativeto which any allocationis Paretodominatedby some allocation for more optimisticbeliefs. Roughly speaking,this is because, when values are common,the agent suffersfrom the principal'stype being bad. Thus when the probabilityof the bad type is high, the agent must be paid correspondinglyhigh compensation(i.e., the principal'swagesare low), implying that the principal'stypeshave low payoffs.Note that with privatevalues,thereis no such thing as pessimisticor optimisticbeliefs since the agent does not care about the principal'stype. 21 Above we describedthe SUPO locus as the outer envelopeof the UPO locus as beliefs vary. Propositions2 and 3 togetherimply that for any beliefs there is a correspondingpoint on that envelope.

394

ERIC MASKIN AND JEAN TIROLE

Proposition3, provedin the Appendix,closely resemblesthe standardline. It is our analog of the First FundamentalTheoremof WelfareEconomics.We also can derivea counterpartto the SecondWelfareTheorem. 4: If ,i is a SUPO allocationfor strictlypositive beliefs ST(that is, it belongs to the intersectionof the SUPO set and the UPO allocations relative to 1i), then ,i is a Walrasianallocation relative to beliefs 'T. PROPOSITION

PROOF:BecauseftJis SUPO,its slackvariablesand ST= Max V,J*(ri*,ci*)

ST solve the program:

such that

{ , r' ,c' }

>v^ forall i 0i*,

(5)

Vz(ic')

(6)

EITir ir<0,

(7)

iCi < o

and

where i* is a type such that w'* > 0 in the UPO programthat ii solves, and where foralli. (8) vi'=EpjV'(`i') I

Let p and -ydenote the Lagrangemultipliersof (6) and (7). Since S =- T solves the above programand, for all i, I'Tis strictlybetween0 and 1, the first-order condition obtainedby differentiatingthe Lagrangianwith respectto g' is: (9)()prAi

r

+

y8=0 yC^i =

O.

The first-orderconditionswith respectto r' and ci imply (10) r' c' =p/y for all i. In view of (9) and (10) and becauseV/(r', ci) is concavein (ri, ci), we infer that c^)}7_} is a Walrasianequilib(ri, c^i)E Di(p, -y).We concludethat {(p, ry),{(r^, Q.E.D. riumrelativeto beliefs s'. COROLLARY: If ,i is a SUPO allocationfor strictlypositive beliefs ST,it satisfies the principal's individualrationalityand incentive constraints:

(I)

-i

EpjVi(Ai )

>

? pjvi(

>,E pVi(

and

(PIC)

J

ij)

h)

i

for all i and h. PROOF:Because,in view of Proposition4, ii is a Walrasianallocation,it must give the type i principal at least the utility she obtains from her "initial

395

PRINCIPAL-AGENT RELATIONSHIP endowment," (ri, ci) = (0, 0). But Vi(0, 0) =

vi3. Hence,

we obtain (PIR). Since all

types of principalhave the same endowment,moreover,they can affordto buy Q.E.D. each other'sequilibriumallocations.Thus(PIC)follows. REMARX:In our definitionof W*, we did not requirethat the principal's utilities exceed the full informationlevels.Nonetheless,the corollaryto Proposition 4 demonstratesthat this propertyholds for all points in W* corresponding to strictly positive beliefs. The corollaryalso vindicates our omission of the principal'sIC constraintsin the definitionof W*.It can be shown,however,that both Proposition 4 and its corollaryare false when beliefs fail to be strictly positive (see Maskin-Tirole(1986)). Given that a competitiveeconomyis sufficientlysmooth,it genericallyhas only finitely many equilibria.For exactly the same reasons, we can draw such a conclusionin our model. PROPOSITION 5: For an open and dense subset of utilityfunctions (satisfying the conditions of Section 2) there exist onlyfinitely many Walrasian equilibria relative to i7r.

The proof of Proposition 5 is standard but uses methods of differential topology that are beyond the scope of this paper. We refer the reader to Mas-Colell(1985) for a comprehensivetreatment. REMARK:Propositions4 and 5 togetherillustratehow muchstrongera concept SUPO is than UPO. For fixedbeliefs thereis a continuumof allocationssolving program UPO. However, generically, only finitely many of these are SUPO. C. Equilibriumin the Principal-AgentGame

We now use our resultsfor the competitiveeconomyto studyperfectBayesian equilibrium of the principal-agentgame. We first demonstratethat one can constructsuch an equilibriumfrom a Walrasianallocation. PROPOSITION 6: There exists a perfect Bayesian equilibriumof the three-stage contract proposal game. More specifically, for any prior beliefs v and any Walrasian allocation for the fictitious economy relative to IT, there exists an equilibrium where all types of principal propose the same contract and where the equilibrium outcome is this Walrasianequilibriumallocation. PROOF: Considera

Walrasianequilibrium{(r, y), { r^,c' }

} relativeto the

prior beliefs 7r. Let i be the corresponding allocation and let v be the vector of

Walrasianpayoffs.From Proposition2, such an equilibriumexists. We first constructthe equilibriumpath of our perfect Bayesianequilibrium. Along the path, all typesof principalproposethe directrevelationmechanism i.Because the agent can infer nothing from this proposal, he does not modify his

396

ERIC MASKIN AND JEAN TIROLE

7T1 for all i. The agent, irrespectiveof his type, accepts the contract in the second stage, and so the principal does not revise her prior probabilities(Pl, P2). Finally, both parties announcetheir types truthfullyin the thirdstage. To demonstratethat this behaviorforms an equilibriumpath, we work backwards from the end. We first show that truthfulrevelationis optimal for both parties in the third stage; next, that it is in the agent's interest to accept the mechanism 2. in the secondstage; and, finally,that for any alternativecontract proposal in the first stage, there exist posterior beliefs and a corresponding continuationequilibriumin which no type of principalis better off than on the equilibriumpath. Becausefi: is a Walrasianallocationrelativeto beliefs vr it satisfiesthe agent's IC constraintsby definition.Hence, if the principalannouncesthe truth in the third stage, the agent will find it worthwhileto do so too if his beliefs are v7. From the corollary to Proposition4, ,ii also satisfies the principal'sIC constraintswhen her beliefs about the agent'stype are (Pl, P2). Hence, truth-telling forms a BayesianNash equilibriumin the third stage, assumingthat the parties have maintainedtheirpriorbeliefs. Becausethe agent obtainsat least the utility iuin the thirdstage,it is optimal for him, given his prior beliefs, to accept the proposal fi: in the second stage regardlessof his type. Hence, the principalwill not updateher priorbeliefs. It remainsto choose off-the-pathstrategiesand beliefs at the first stage that deter the principalfrom proposinga contractother than ,u:. Suppose that the principal proposes some other, finite mechanismm. Because this proposal is never made in equilibrium,beliefs X are not determinedby Bayes'rule. Instead, they can be arbitrary.For each possible vector of beliefs, there is at least one

prior beliefs about the principal's information, i.e., 'i =

corresponding continuation equilibrium (see Kreps-Wilson (1982)).22 Let Vm be

the convex hull of the set of continuationequilibriumpayoff vectors (for the principal) corresponding to m. For any posterior beliefs 9', let Vm4(7) be the set of

continuation equilibriumpayoff vectors for the principalwhen m is proposed and beliefs are 'T.If we supposethat,in the case of multipleequilibria,a random public event (e.g., sunspots)makes the selection, then Vme(r)is a convex and compact subset of Vm. For payoffvectors v E Vmand beliefs STdefinethe correspondence (11)

(0s7rv)

I E-argmax E s (7717T

i( vi -V vi) x

(s)

where vbiis the type i principal'sWalrasianpayoff.Correspondence (11) is closely analogous to the well-knownDebreu mapping used to establish existence of competitive equilibrium.The cross product of the belief and payoff sets is compact and convex, and the correspondenceis upper hemicontinuous(see 22 It need not be the case that eitheror both types of agent accept the proposal m in a given continuationequilibrium.But, whetheror not the proposalis accepted,thereis still an equilibrium payoff.

PRINCIPAL-AGENT RELATIONSHIP

397

footnotes 15 and 16) and convex-valued(see footnote 17). Hence, it admits a fixed point, (go, vo). Assume first that (12) gvi( V-Oi) > 0. J is not empty, because v^is SUPO. and J={ijv0Y}. Let I={ijv0>Y} maximizes From (12) I is not empty. BecauseT7o Ev2r(v'- v^), v7i= 0 for i e J. as be written can Thus, the agent IR and IC constraints (13)

E

ror< O

is,

and ci < 0

(14) is,

where roiand c' denote the negativeslackvariablesassociatedwith vo. Now, each type i in I prefersv' to her competitivepayoff,which means she cannot afford rdiand c' at competitiveprices p and 'y: + -Ci > O, for i E-I. (15) Aroi But (13) through(15) are clearlyinconsistent.Hence,(12) is impossible. We conclude that _2 (v would exist ST such that 7T

<-^i) -

0. Thus, for all i, v < v^ (otherwise there 0, a contradiction). Because v0 E go

Yv^)>

7TOand

vo constitutebeliefs and correspondingcontinuationequilibriumpayoffs that no type of principalprefersto her Walrasianpayoff. Hence if we assign 'go and vo to m, no type of principal will deviate from proposing ii:.

Q.E.D.

To summarizethe constructionof the proof, each type of principalproposes the Walrasianallocation AI as a directrevelationmechanism.In equilibrium,the agent accepts the proposal,and, in the the thirdstage, the two partiesannounce their types truthfully.Shouldthe principalpropose some other mechanism,the agent's beliefs and the continuationequilibriumare chosen so that all types of principalare no betteroff than with ii:. That this is possibleis particularlyclear mechawhen n = 2. Supposethat the principalproposessome out-of-equilibrium nism m. If the agent attachesprobability1 to a = al, then the type 1 principal can derive no more utility than U0,which is clearly less than that which she derives from #.. Similarly,the type 2 principalobtainsless utility from m if the agent believes a = a2 than from jr..From continuityand because1i: is SUPO, there exist intermediatebeliefs whereboth types are no better off with m than with at:(see Figure2). REMARK: Proposition6 is a reflectionof the idea that, far from therebeing a conflict among the differenttypes of principal,they mutually gain from the agent'sincompleteinformation.They take advantageof this incompleteinformation by revealing no informationuntil their proposal is accepted and then exploitingthe fact that the agent'sconstraintsneed hold only in expectation.

398

ERIC MASKIN AND JEAN TIROLE v2.

SUPOfrontier Payoff s if

(VI (' 1),V2Q(2)) Locus of payoffs corresponding to m VI

(v1,V2)

KPayoffs if 7; 2= I FIGURE 2.-Existence

of a PBE corresponding to a Walrasian equilibrium.

D. Uniqueness

We provedin subsection3C that thereexists an equilibriumof the three-stage principal-agentgame.This equilibriumcorrespondsto a Walrasianallocationof the fictitiouscompetitiveeconomy.We now investigateuniqueness. Sequentialgamesof incompleteinformationare often plaguedby a plethoraof equilibria.One may wonderwhethersuch is the case here. Can any strongUPO allocationbe an equilibriumoutcomeof the three-stagegame?Do thereexist any suboptimal equilibria?As we shall see, the answerto both questionsis "no." Indeed, we demonstratethat only Walrasianallocations relative to the prior beliefs S can arise as equilibriaof the principal-agentgame. Since such allocations are, generically,locally unique, the same is, therefore,true of the game's equilibriumoutcomes. To establishour uniquenessresult,we expandthe class of permissiblemechanisms somewhat. In particular,we now include mechanismsin which a third party, in addition to the principaland agent, chooses from a set of messages. Moreover,ratherthanjust dealingwith finitemechanisms,we let the permissible set M* include all mechanismsm such that, if the principal'sbeliefs about the agent at the time the mechanismis to be played are given by the prior beliefs (P1lP2), (a) there exists a perfect Bayesianequilibriumof m regardlessof the agent's beliefs about a; (b) for any SUPO payoff vector, (61,...,

v6), there exists,

for some vector of agent's beliefs s', an equilibriumof m for which the equilibriumpayoffs, (v i,..., vn), satisfy vi< v^for all i. Conditions(a) and (b) are admittedlytechnicalbut expressthe naturalrequirementsthat (i) the principal should be able to predictthe outcomeof her proposal(equilibriumshould exist) and (ii) equilibriumshouldbe well-behavedas a functionof beliefs ((b) is satisfiedif the equilibriumpayoff correspondenceis upper hemicontinuousand convex valued). The conditions,moreover,are automaticallysatisfiedfor finite mechanisms.As we noted in the proof of Proposition6, (a) is guaranteedfor finite mechanismsby sequentiality(see Kreps-Wilson(1982)). Condition(b) for finite mechanismswas establishedin the proof of Proposition6. Indeed,one can

PRINCIPAL-AGENT RELATIONSHIP

399

easily confirmthat conditions(a) and (b) werethe only propertiesof mechanisms requiredfor demonstratingthe existenceof equilibriumin our principal-agent game. Hence, Proposition6 continuesto hold for the largerclass M*. Besides enlargingthe class of availablemechanisms,we also strengthenour assumptionsabout the agent's utility function. Specifically,we suppose that it satisfiesa conventional"sorting"assumption. ASSUMPTION

2:

U(Y1 t, 61)

U(Y,t,61i)

(Uy(Y' t,

02)

for all (y, t).

Ut(Y,t,602)

PROPOSITION 7: Let M* be the class of admissible mechanisms. Then, given Assumption 2 (in addition to the assumptions of Section 2), any perfect Bayesian equilibrium of the principal-agent game is Strong UnconstrainedPareto Optimal (SUPO).

That equilibriumallocationsmustbe Paretooptimalrelieson the abilityof the principalto break an inefficientequilibriumby proposingan alternativemechanism that, whateverthe agent'sbeliefs turnout to be, makes(at least) one of her types better off. This "equilibriumbreaking"can be accomplishedby the following simple mechanismm*. First, the principaland agent announceprobability vectors g and gTa(correspondingto the agent'sbeliefs about the principal'stype when m* is proposed).If s7 * S7a the null contractis imposed.If s7 = 7a = ST, the principaland agent play the Walrasiandirect-revelationgame correspondingto 7T. I.e., they announce their types simultaneously,and the outcome is the Walrasianallocation for the announcedtypes relativeto s7 (the game must be somewhatmodifiedif therearemultipleWalrasianequilibria).Notice that it is an equilibriumof this gamefor the two playersto announcethe agent'strue beliefs and then announce their true types. This equilibrium,therefore,is Walrasian relativeto the agent'struebeliefs,and so does the trickof equilibriumbreaking. The weaknessof m* is that, althoughthe above "truthful"equilibriummay be particularlysalient, there are other, "perverse"equilibriaof m* in which the players(a) announcedifferentbeliefs,or (b) announcethe samebut false beliefs, or (c) announce their types falsely. The proof of Proposition7 (see Appendix) constructsa more elaboratemechanism,based on m*, in which these perverse equilibriaare eliminatedand only the Walrasianoutcomeremains. The Pareto optimalityof equilibriumdependsimportantlyon privatevalues. As we noted followingProposition3, SUPO allocationsdo not even exist relative to all beliefs in common-valuesmodels such as the Spencian labor market. Moreover,even for beliefs relativeto which a SUPO allocationdoes exist, there can be many inefficientequilibriaeven if the principaluses the sort of mechanisms invoked in the proof of Proposition7. This is because to break an inefficient equilibrium,as we have noted, the principal needs to propose a mechanismthat, regardlessof the agent'sbeliefs, is better for one of her types.

400

ERICMASKINAND JEANTIROLE

But with common values and "pessimistic"beliefs by the agent (beliefs that attach high probabilityto the bad type(s) of principal),all the principal'stypes may actually be worse off than in the inefficientequilibrium.Thus the equilibrium cannot be broken. One can interpretProposition7 as an illustrationof the idea that if, relativeto beliefs, thereare gains fromtrade,the principalought to be able to exploit them. The common-valuesmodel is not a counterexampleto this pnrnciplebecause there, if the principaltries to overcomethe inefficiency,the agent'sbeliefs may change in such a way that there are no gains from trade. PROPOSITION8: Given the hypotheses of Proposition 7, any perfect Bayesian equilibrium allocation ji:. of the three-stagegame is a Walrasian allocation relative to prior beliefs iT. PROOF: From Proposition7, ,t is SUPO . Hence, becauseit satisfiesthe IR and IC constraints of program (UPO) for beliefs 7T, it is SUPO for iT. By assumption, iT is strictlypositive, and so, from Proposition4, ,Iu is Walrasian

relativeto v.

Q.E.D.

We noted above that any equilibriumallocationcan be thoughtof as arising from a pooling equilibrium,in which all types of principalpropose the same mechanism.Proposition8 demonstratesthat,in general,somepoolingis essential in equilibrium.A WalrasianallocationgenericallystrictlyParetodominatesthe full-informationpayoff vector. Thus the fact that the equilibriumallocationis necessarilyWalrasianimpliesthat the principalcannot perfectlyrevealher type by her proposal. We know from Proposition5 that the Walrasianequilibriaof the fictitious economy are genericallyfinite in number. In view of Proposition 8, we can conclude the same for the perfectBayesianequilibriaof our three-stagegame. PROPOSITION9: For an open and dense set of utility functions (satisfying the hypotheses of Proposition 7), there exist only finitely many perfect Bayesian equilibriumallocations of our principal-agentgame.

Propositions7 through9 are obtainedby extendingthe class of mechanisms beyond DRM's. Another,and quitedifferentrouteto efficiencyand uniquenessis to refine the concept of perfect Bayesianequilibrium.Specifically,even if the principalis constrainedto proposeonly allocations(i.e., DRM's),the application of the Farrell(1985) Grossman-Perry again rules out (1986) (FGP) refinement23 23 In our context, this refinementrequiresthat theredoes not exist a subset of types S and an alternativeallocation such that types in S (weakly)gain (and the other types lose) relative to equilibriumand the allocationsatisfiesthe agent'sIC and IR constraintsif the agentupdateshis prior needed to have probabilitiessum to 1, the posterior beliefs so that, ignoringthe renormalization probabilityof a type who strictlygainsis the sameas the priorprobability,thatof a typewho strictly loses is zero, and that of a typewho is indifferentis intermediate.

401

PRINCIPAL-AGENT RELATIONSHIP

all but SUPO allocationsin equilibrium.To continueour Walrasianmetaphor,a roughintuition for this resultis that the core coincideswith the set of Walrasian allocations(in large economies).Thus, if an allocationis Walrasian,thereis no subset of principal'stypes that can makethemselvesbetteroff by tradingamong themselves.Conversely,if an allocationis not Walrasian,there does exist such a coalition. PROPOSITION10: Walrasianallocations relative to prior beliefs STsatisfy the FGP refinement. Conversely, if either n = 2 or there exists a unique Walrasian equilibrium for any 7T, any perfect Bayesian equilibriumallocation ,(1:of the three-stage game (where the principalproposes DRM's) that satistfies the FGP refinementis a Walrasian allocation relative to prior beliefs 7T.

PROOF:See the Appendix. 4. QUASI-LINEARUTILITIES

Much of the incentives literatureconcerns the special case of quasi-linear objectivefunctionsfor the principaland the agent: Vi

f(y)

_ t

(=1,.,n),

and Uj= t- 4'(y)

(i=1,2).

For our purposes, the most importantfeature of these functions is that the shadow values of the two constraintsin the full-informationprogramare independent of the principal'stype.24That is, the marginalrate of substitution between the two slack variablesis the same for any type; and so there are no gains to be reaped from trade. The Walrasianequilibriumof the fictitious competitive economy is autarky. Hence, Proposition 1 does not pertain to quasi-linearutilities. Indeed,from previousanalysis,we immediatelyobtain the followingproposition. PROPOSITION11: In the quasi-linearcase, the uniqueequilibriumpayoff vector of the three-stage game is the full-informationvector v.

Proposition 11 asserts that, with quasi-linearutilities, the principal neither gains nor loses if her type is revealedto the agentbefore the game is played. Of course, this is an outcomeof the nongenericnatureof the quasi-linearcase. 24

Becausethe payofffunctionsare linearin transfers,we mightas well assumethat transfersin a solution to (Fi) are deterministic; we can replaceany randomtransferby its meanwithoutaffecting anything.Now, formingthe Lagrangianfor (F') and optimizingwith respectto the transfersimplies that p' = 1 and y' = Pi regardlessof the value of i.

402

ERIC MASKIN AND JEAN TIROLE 5. SUMMARY

When values are private,the principalstrictlygains, in general,by concealing her type until the contractshe proposesis carriedout. This concealmentenables her to be constrainedby the agent'sindividualrationalityand incentivecompatibility constraintsmerelyin expectation,ratherthan type by type. One can, in fact, view the differenttypes of principal as competitivetradersin the slack variablesassociatedwith these constraints;one trader'sviolationof a constraint is counterbalancedby anothertrader'sacceptingsome slack.In fact, the equilibria of the three-stageprincipal-agentgame correspondexactly to the Walrasian allocationsof this competitiveeconomy(and so, in particular,they are efficientin a strong sense). The Walrasianinterpretationis illuminatingin severalrespects.As we have just indicated,it helps us understandwhy the principalgains from pooling and how she profits from the agent'signoranceof her type. It also explainswhy, in equilibrium, the principal's own incentive compatibility constraints are not binding. Just as consumerstradingfrom equal endowmentsdo not envy each other's allocationsin Walrasianequilibrium,so no type of principalprefersthe equilibriumallocationof some other type. The analogywith Walrasianequilibrium,however,relies on the privatenessof values and the absenceof moralhazard.We have alreadynoted that, in common value models, inefficient(and hence non-Walrasian)equilibriamay exist in large numbers (see Maskin-Tirole(1988) for greater elaboration).In such models, unlike that of this paper,it is no longertrue that, withoutloss, the principalcan postpone revealing her type until the third stage. She may wish to disclose informationabout herselfin orderto influencethe agent'saction. Her proposal must thereforestrike some balancebetween total disclosureand completeconcealment. Although our model is alreadyquite general,many of our assumptionscan be relaxed further. The two crucial assumptionsfor our results are that (i) the principal's informationparameterdoes not enter the agent's utility function (thereby avoiding signalingphenomena)and (ii) the full-informationprogram includes at least two bindingagent constraints(so that the principal'stypes are able to trade slack variables).Thus our results would not be affected by (a) multidimensionaltype and action spaces;(b) nonmonotonicutility functions; (c) one-sided common values, in which the principal'sutility depends on the agent's information(indeed, in public sector applications,where the principal acts on behalf of society (e.g., the public good or regulationexamples),her objectivefunction may take accountof the agent'swelfare;in that case, V is a function of 0 as well as of (y, t, a); (d) statisticaldependencebetweena and 0 (in this case, parties'expectationsmustbe madeconditionalon theirown types); (e) arbitrarynumber of agent's types (focusing on two types allowed us to simplify exposition since only two constraints-one IR and one IC-were binding; what matters is that there be at least two binding constraints); (f) reservationutilitiesthat dependon the agent'stype (for the same reason).

PRINCIPAL-AGENT RELATIONSHIP

403

None of the generalizationsrequiresfurtherargument;they demandonly more involvednotation. Department of Economics, Harvard University,Cambridge,MA 02138, U.S.A. and Department of Economics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. ManuscriptreceivedMay, 1986; final revision received March, 1989.

APPENDIX OFLEMMA PROOF 1: To see thatthe IC constraintfor the type2 agentholds,considerthe program (UPO). We will show that a solutionto this programsatisfiesthe deletedconstraint.Observefirst that, if ,u: is a solutionto the program(UPO),then

W

2

Formula(*) holds because,were it violated,the (pooling)allocation,u definedso that, for all i, A' = #' = A' would satisfythe constraintsof the program(UPO) and generatea highervalueof the maximandthan ,u:.Now, if A. violatesthe deletedIC constraint(i.e., the type2 agentstrictlyprefers p4 to A'?),let us define ii so that,for all i, XI= I'2= A4.The allocationp4satisfiesthe constraintsof the program(UPO) and, from(*), generatesat least as high a valueof the maximandas ,u. Indeed, because the type 2 agent strictlyprefers,u; to A' we can slightlydecreasethe transferfrom the principalto the agent in p4(Assumption1 guaranteesthat such an increaseis possible)without violatingthe constraints.But then ,d generatesa highervalueof the maximandthan ,u, a violation of p4'soptimality. Q.E.D. PROOFOF PROPOSITION 1: Consider the solution (j.,

p',

y') to (F'), where p' and y' are the

shadowpricesof the IR' and IC' constraints.For any two typesof principal,say 1 and 2, it can be shown that, for almost all choices of utility functions, V1 and V2, satisfying the Section 2 shadowpricessatisfyp'/-yl * p2/y2.25 assumptions,the corresponding For an arbitrary (random) allocation 1/., let r'(p.)

and c'(p'.) be the negatives of the slack

variablesassociatedwith the IR' and IC' constraints: r'(pA-) 3U- U2(t2)

and c'(p4) 3Ul(

-Ul(4

= 0 and c(ji.) = 0. Moreover,for beliefsso,the constraintsof the (UPO)program In particular,r%(K.) can be expressedas

le.r'(p4)S0

IR: l

and IC: 5?1clc(p.) <0. l

Thus, to satisfythe constraintsof (UPO),the negativesof the slackvariablesneedonly be nonpositive on average,and not for eachvalueof i individually. 25"Almostall" means"for an open and densesubset."AndreuMas-Colellhas providedus with a proof of this assertion.

404

ERIC MASKIN AND JEAN TIROLE

Consider the following perturbed version of the full-information program (F'): (Max EpJ V'( C, ) (F*)

i- r ,

U2(L2 Ul (K)> 1

suchthat

Ul(!2 ) - C'.

By definition of the shadow prices p' and -', the maximized value, v', of the maximand approximately equals v' + p'r' + y'c', for small values of r' and c'. Let ?., be the solution to F*'. Choose negative slack variables (rl, cl) for the type 1 principal; define negative slack variables c2 = for the type 2 principal; and take {r' = 0, c' = 0} for types 3 'T/s2)c'} {r2 = -(&T/12)r', through n. Via program (F*), we obtain new maximized utilities for the type 1 and 2 principals (the other types' payoffs are the same as under full information): Vl* Vl - pr

(al)

+

yl

and (a2)

If p1/-1 (a3)

v*

*

-

2-

r

T

Y

(as is generically the case) one can choose (rl, cl) small enough26 that

2

vl*- P > 0

and

v*2_ 2>O.

For instance, if 1/yl > p2/y2, so that the IR constraint is relatively more costly for the type 1 principal, she can "accept" some slack on the IC constraint in exchange for being permitted "negative slack" on the IR constraint. That is, rl is positive and cl negative. From our choice of slack variables, the allocation ,u* satisfies the constraints of program (UPO). Thus, in view of (a3), ,. Q.E.D. is not unconstrained Pareto optimal. PROOF OF PROPOSITION3: Let (p, y) and {(r', c')}7 1% be a Walrasian equilibrium relative to beliefs ST and let ,u be the corresponding allocation. If this allocation is not SUPO, then there exists a UPO allocation &: for beliefs ST such that, for all i,

(bl)

EP, v(,

E P, ,(gj'), J

J

where inequality (bl) is strict for some i and, if ST is not strictly positive, strict for all i. Let { P', ='}I. 1 be the negative slack variables associated with the allocation a.. Then, from the IR and IC constraints of program (UPO), ?

7?T? SO0 and

(b2) ,=1

~~~~~n 0. SO<

ET ,=1

By definition of the Walrasian equilibrium and (bl), (b3)

pi?+yc`>pr'+yc',

foralli,

where the inequality is strict for some i (all i if 'T is not strictly positive). Multiplying (b3) by 7T', summing over i and recalling that the right-hand side of (b3) is zero for all i, we conclude that either Q.E.D. E2'r' >0 or E "Ic > 0, a contradiction of (b2). PROOF OF PROPOSITION7: Let us assume for convenience that n = 2 (the argument extends to any number of types). Suppose contrary to the proposition that there exists an equilibrium with payoffs v = (v1, v2) that are not SUPO. For each vector of beliefs s' and for each corresponding Walrasian allocation ,u (7'T),

(cl)

there exists i such that Ep Vi ( L())I

> v'.

I

26 The reason for choosing rl and cl small is to ensure that the approximations (al) and (a2) are good enough for (a3) to hold.

405

PRINCIPAL-AGENT RELATIONSHIP Moreover, generically, we have pIV'(u. (ST)) >

(c2)

pjV'(

for all i*h

(S))

and either

(c3)

U2( u2()

or

(c4) (c)

U2 ( t12 ( Iq ))
Withoutloss of generality,assumethat (c4) holds.Then U2(u()

(cS)

IT

( from (c4 ) and E 'U2

u

'2() ='

(

u)

For each strictlypositivevectorST and Walrasianallocation,U(Ir), choose E> 0 sufficientlysmall so that there exists a slight perturbation ,ii(qT) such that (cl), (c2), (c4), (c5) remain true, and also

(c6)

(c7)

T

I

u,

EqTU2(X2(qT))>

(i=1,2),

>E

EqZU2(X52(T))

?

IU2(#l(qT))

and IT tUl

(c8)

I( SIT

IT qt'L (2

)

IT)

If I'T= 1, then choose E and pri(Ir) to satisfy the same conditionsexcept (a) drop the left-hand inequalityin (c6) for i =j and (b) for j = 2, impose U1(, i(A(2)) = U1(j22(1r))(insteadof (c8)) and U1(gi'j(4)) > U1(A2'2(4)) (see the derivation of (c25) below for why we can impose this last inequality). Condition (c6) ensures that ,U(1Q) Pareto-dominates t'Li(12) by at least E (in utility terms).

Conditions (c7) and (c8) requirethat the type 2 agent strictly prefershis perturbedWalrasian allocationto his reservationutilityand that each type strictlyprefershis own perturbedallocationto that of the other type. That a perturbationof ,u(I*) can be found satisfying(c7) and (c8) is an immediateconsequenceof the sortingcondition.That such a perturbationcan also satisfy (cl) through(c6) (except(c3)) followsfromcontinuity. We must make sure that the mechanismwe constructsatisfiescondition(b) of the definitionof M*. To this end, choose a countabledense subset{ v(1), v(2),... } of the set of SUPOpayoffvectors (such a selectionwould be unnecessaryif we did not have to satisfy (b)). Because,generically,the SUPO allocations associatedwith given beliefs ST are locally unique, we can choose the subset v(1), v(2),... } so that the corresponding beliefs { iT(1), iT(2),... } are all distinct. For t = 1, 2,..., let be the Walrasian allocation associated with v(t) and let *i(iT(t)) be the corresponding perturbed Walrasian allocation satisfying (cl) through (c8) (except (c3)) above. Define

tL.(iT(t))

I (T)

if 'T= IT( t)

I'T (s)(cl)

otherwise, where '1(S'T)is a perturbation (satisfying - (c8) (except (c3)) of an arbitrary Walrasian allocation,u(1r).

,d.(qIT)

(c9)

I

k For each I', choose

such that

3

(cl0a)

ul( i'3) = Ul

(cl0b) Un( Ud(

2) >

(cll) (c12

2),

U2(A!2) > U2(9i3) E

(i1,

'iUJ( K) > E, "tUJA'),

and

vl(133

V1M

- VI(l2)

> Vl{M _#

ll2

2),

ERICMASKINAND JEANTIROLE

406

That we can satisfy(c1O)through(c12)is a directimplicationof continuityand the sortingcondition. Formula(c13) followsif 23 entailsa largeenoughmonetarypayment(whichis simplythrownaway) by the principal.Thereis no contradictionbetween(clOb)and (c13) becausewhat the agentreceives need not equal what the principalpays (we are slightly abusing notation by writing both the " principal'sand agent'sutilityas a functionof the sameallocationI2i). Similarly,take so that (cl4a)

U2(4)>

(cl4b)

U2(!4)

(clS)

UJAX) ' Ul(#!4)

(c16)

U2( i1) 2

=

(i= 1, 2),

> ', U2(#!4),

r'U2(#!2)

and (c1 7)

V (iv2( 2) - V2(y4) > V2(,i)

Finally, choose

(c18)

'5

-

V2(

so that (i= 1,2),

U2(,i5) >'

(c19)

IhTS (ih)>

(i

)

(i,j=j

1,2),

h

(c20)

1, 2),

Ul ( ,i12) > Ul (A=5)(

and (c21 )

v2( o2)

-

V2(gj)>

V2(,22)-V2(4

To satisfy (c18) through(c20), we can choose #5 so that y = 0 and t is slightlypositive (thus Uj(

!5) - U,i,i, j,

=

1,2). To ensure (c21) we can, as above, require a large monetary payment by the

principalshould she set a = al. Considerthe followingcontractm*. In this contract,a thirdparty first announcesa vector of "beliefs," ', which, in equilibrium,will turn out to be (at least approximately)the agent'sbeliefs about about the principal'stype.Theprincipaland the agentthenmakesimultaneousannouncements their types. That is, the principalannounces 'E {a1,a2}. Becausewe have added three more agent's announcement 6 hes in "types" for the agent-corresponding to "i, ", and --the contract specifies the allocation the and = are the announcements If a', = 0J, }. { 02, 03, 04, J x,, & (Xf)(definedby (c9)). Moreover,if j]e {1,2}, the thirdpartyis given a (small)monetarypayoff &ji (such a payoff is feasible since #./(s') does not quite attain the Walrasianallocationfor i') and nothingif j E {3,4,5}. We shall argue that, if the principalproposes m*, the agent will accept it. There exists an equilibriumof m* in which the thirdpartyannouncesST equal to the agent'strue beliefs s7 and in which the principaland agentboth announcetheirtypestruthfully.Moreover,in any equilibriumof m* the principal and agent are truthful.Therefore,the only possible allocationsresultingfrom proposing m* are the i:(si). But (cl) implies that, for any s', there exists at least one type i of Thusnon-SUPOequilibriumallocationsare impossible. to v'. principalwho prefers#'.(7'r) ' ' and truthtellingby the principal and agent constitute an We first demonstratethat = equilibrium.Notice that, if ST= ST and the principalis truthful,(c7), (c8), (cll), (c12), (c15), (c16), (c19), and (c20) imply that the agent is truthful.Moreover,(c2) guaranteesthat the principalis truthfulif the agent is. Now, if the thirdparty announcesS'T= ST and the principaland agent are truthful,the thirdparty'spayoffis maximal,sincethe probabilitythat the agentannounces01or 02 is he couldmakethatcould possiblyraisehis payoff. one. Hence, thereis no otherannouncement We next show that, if 'T = 4, the only possibleequilibriumis the truthfulone. Suppose,to the contrary, that there is an untruthfulequilibrium.In this equilibrium,let w*', i = 1,2, be the probabilitythat &= a'. Now, if (7T1, *2)= (qT 2), thenthe argumentfromthe precedingparagraph impliesthat the agentis truthful,whichin turnimpliesthat the principalis truthful,a contradiction. Hence ( 4, T*) (71 2 ) > CASEI: iT*2,

and X1< 7.

PRINCIPAL-AGENT RELATIONSHIP

407

Note first that (cl1) implies that the type 2 agent does not set 6=03. Now, if V2(,i) > V2(g2), then in view of (C5), the allocation 2 satisfies the agent's IR and (trivially) IC constraints and yet, from (c6), generates more utility for the type 2 principal than v2, an impossibility. Hence, V2(,U2) > V2( 22)

(2)

If Y-T*U2(#,l) > YT*U2(f'2),

then, in view of 7T > 7T and the second inequality

of (c7), U2(g)

U2( i ). But if this last inequality held, then, because of (c22), ,12 would satisfy the agent's IR and (trivially) IC constraints but generate a higher payoff than v2 for the type 2 principal, an impossibility. We conclude that U2( ,'l ) < U2( A'22

(2) 7T*'U2(

)
T*U2(,U1),

and therefore,

that the type 2 agent does not set 0 = 61. In turn, (c23)

implies, together with (c14b) and (c16), that the type 2 agent does not set 0=64. From (cl1), he does * not take = 063. Finally, (c4) and (C5) imply that U2(I ) > U2(,'2), and so 7*> and (c19) imply that the type 2 agent does not set = 85. We conclude that the type 2 agent sets =2 < 7T, the type 1 agent does not set 0= 2. From (c15) he From (clOa) and (cl0b) and the fact 4T* does not announce 0=64. Finally, (c20) implies that he does not set = 65. Hence, the type 1 agent must either set 6= 61 or 0= 3 (or randomize between them). In the former case, given that the type 2 agent is truthful, the type 1 principal is clearly better off announcing &= al. But (c13) implies that she also is better off with &= a' in the latter case (and hence for any randomization between 01 < 7T'. and 03). Thus, in equilibrium, the type 1 principal takes &= al, contradicting 4T* CASE I:

*T7 <

2and * >&.

Note first that, if T2 < 1, (2)

U1( ,i2 ) < U1( i2 );

otherwise, in view of (C5), the allocation IL. satisfies the agent's IC and IR constraints. But ,. gives the type 2 principal more utility than v2, a contradiction. Now, (c24) and (c8) imply that (C25)

U1(

X

) > U1(,U2)

for 7r2 < 1. Now, from our choice of ,U(7r) (see the passage following (c8)), (c24) becomes an equation and (c25) is satisfied even when r2 = 1. In view of (c8), (c24) (both when T2 < 1 and T2- 1), and (c25), 47*> &1 implies that

(C26)l ETU1( IU ) > DrSTU1 ( '2), and so the type 1 agent will not set 6= 02. This, in turn, together with (clOa), (cl0b), and (c12), implies that he will not announce 6 = 03. Now, (c15) implies that the type 1 agent will not set 0=04. Finally (c20) implies that he will not announce 6==5. We conclude that (c27)

the type 1 agent must announce 6 = 61.

< 1, the type 2 agent will not announce 6= 1, and From (cl4a) and (c14b) and the fact that 7T*2 from (cl1) he will not announce 6= 03. Hence, his announcement must be 62, 04, or 65 (or some randomization among them). Now, if he announces 02, then, from (c27) and (c2), it is optimal for the type 2 principal to announce &= a2. But from (c17) and (c21) the same is true for announcement 64 or 05. Hence, a = a2 must be the type 2 principal's equilibrium announcement, contradicting < 72 We conclude that, if ST = 4, only a truthful equilibrium is possible. We next observe that in any equilibrium where the agent never makes an announcement other than 61 or 02, both the principal and agent are truthful. To see this, note that if in equilibrium the type 1 agent announces J (=01, 02}, then (clOa) and (cl0b) rule out the choice 6= 02 unless Tl= 1, in which case (c25) rules out this choice. Moreover, if the type 2 agent announces 6 {01, 0 02), then (cl4a) and (cl4b) imply that 0=02, unless 752= 1, in which case (c23) ensures 6= 02. Hence, the agent is truthful in equilibrium, which in turn implies that the principal must be. Now, the third party maximizes his payoff by maximizing the probability that the agent announces 6 G { 61, 02). But, as we have seen, the party can ensure that the probability is 1 by setting ST= 4. This

408

ERIC MASKIN AND JEAN TIROLE

does not imply that the third party does set ST = ST in equilibrium because the truthtelling constraints continue to hold for S' somewhat different from *r. But it does mean that S' is approximately Ir. In any case, the only possible equilibria are those where the agent never makes an announcement other than 01 or 02. But, as we have shown, this implies that the principal and agent are truthful. Hence, any Note finally that (c18) implies that if the principal equilibrium allocation is in the class , proposes m*, the agent will accept it. The fact that there always exists an equilibrium of m* (where 'T= ST and the principal and agent are truthful) implies that m* satisfies condition (a) of the set of admissible mechanisms M*. To see that it satisfies condition (b), consider SUPO payoffs ( 'I, 12). If ( 'I, v2) = v(t) for some t, then (c6) implies that for beliefs iT(t), (0, v62) Pareto-dominates the truthful equilibrium payoffs for m*. Moreover as long as, for some t, I' - vI(t)I < (e/2) for all i, (c6) implies the same conclusion. Now, this last inequality is satisfied because the v(t)'s are dense in the set of SUPO payoffs. Hence, Q.E.D. condition (b) is satisfied, and m* belongs to M*. The arguments in the proof of Proposition 7 are somewhat involved, but the mechanism m* is quite simple. The third party first announces the agent's beliefs about the principal's type and then the principal and agent announce their types. The allocations p', 4i, and t; simply ensure that, in equilibrium, the principal and agent announce their types truthfully if the third party announces the agent's true beliefs. PROOF OF PROPOSITION10: That a Walrasian allocation relative to IT is an FGP equilibrium is trivial and results from the fact that Walrasian equilibria belong to the core. To prove the converse, let v* denote an FGP equilibrium payoff vector and consider the correspondence from the set of beliefs ST and feasible payoffs v into itself:

(1,V)v

(

ifI' (v' - V'*)k> O for all i,

-max[O, vJ- vJ*]kS

vJ-vJ*]k -max[O, J

foralliand

j,

where k = max[O, max (vh - vh*)] } x

{ jj i

is the Walrasian payoff vector relative to

ST}

By construction, a fixed point of this nonempty, upper hemicontinuous and convex-valued correspondence puts zero weight on types who "lose" relative to the equilibrium (v' < v'*), and preserves for types who are strictly better relative weights with respect to prior beliefs (i.e., (1/7iT') = (iJ/1J) off (v > v'* and VJ > vJ*)). (Note also that Fi/7Tr < J/lrJ if v' = VI*and VJ>VJ*.) If, for this fixed point, v' < v'* for all i, then the equilibrium payoff vector is Walrasian. If there exists i such that v > v' then by construction k > 0 and the equilibrium payoff vector v* does not satisfy the FGP Q.E.D. refinement. REFERENCES ADMATI, A., AND M. PERRY(1986): "Strategic Delay in Bargaining," mimeo. AUMANN, R. (1974): "Subjectivity and Correlation in Randomized Strategies," Journal of Mathematical Economics, 1, 67-96. BARON, D., AND R. MYERSON (1982): "Regulating a Monopolist with Unknown Costs," Econometrica, 50, 911-930. CRAWFORD, V. (1985): "Efficient and Durable Decision Rules: A Reformulation," Econometrica, 53, 817-837. DASGUPTA, P., P. HAMMOND,AND E. MASKIN (1979): "The Implementation of Social Choice Rules," Review of Economic Studies, 46, 185-216. DEBREU, G. (1959): The Theoryof Value. New Haven: Yale University Press. FARRELL, J. (1985): "Credible Neologisms in Games of Communication," mimeo, MIT.

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FUDENBERG,D., AND J. TIROLE(1983): "Sequential Bargaining with Incomplete Information," Review of Economic Studies, 50, 221-247. (1989): "Perfect Bayesian and Sequential Equilibria," mimeo, MIT. GREEN, J., AND J.-J. LAFFONT(1979): Incentives and Public Decision-Making. Amsterdam: North-Holland. GROSSMAN,S., AND 0. HART(1981): "Implicit Contracts, Moral Hazard, and Unemployment," American Economic Review Papers and Proceedings, 71, 301-307. GROSSMAN, S., ANDM. PERRY(1986): "Perfect Sequential Equilibrium," Journal of Economic Theory, 39, 97-119. GUESNERIE, R., AND J.-J. LAFFONT (1984): "A Complete Solution to a Class of Principal-Agent Problems with an Application to the Control of a Self-Managed Firm," Journal of Public Economics, 25, 329-369. J. (1971): "The Private and Social Value of Information and the Reward to Inventive HIRSHLEIFER, Activity," American Economic Review, 61, 561-574. KREPS,D., AND R. WILSON(1982): "Sequential Equilibrium," Econometrica, 50, 863-894. LAFFONT,J.-J., AND E. MASKIN(1982): "The Theory of Incentives: An Overview," in Advances in Economic Theory, ed. by W. Hildenbrand. Cambridge: Cambridge University Press, 31-94. LAFFONT,J.-J., AND J. TIROLE(1986): "Using Cost Observation to Regulate Firms," Journal of Political Economy, 94, 614-641. MAS-COLELL, A. (1985): The Theoryof General Economic Equilibrium:A Differential Approach. New York: Cambridge University Press. MASKIN,E., AND J. RILEY(1984): "Monopoly with Incomplete Information." Rand Journal of Economics, 15, 171-196. MASKIN,E., AND J. TIROLE(1986): "Principals with Private Information, I: Independent Values," Harvard University Discussion Paper .# 1234. (1988): "The Principal-Agent Relationship with an Informed Principal, II: Common Values," mimeo, M.I.T. MIRRLEES, J. (1971): "An Exploration in the Theory of Optimum Income Taxation," Review of Economic Studies, 38, 175-208. MUSSA,M., ANDS. ROSEN(1979): "Monopoly and Product Quality," Journal of Economic Theory,18, 301-317. MYERSON,R. (1979): "Incentive Compatibility and the Bargaining Problem." Econometrica, 47, 61-73. (1983): "Mechanism Design by an Informed Principal," Econometrica, 51, 1767-1798. SOBEL,J., AND I. TAKAHASHI (1983): "A Multi-Stage Model of Bargaining," Review of Economic Studies, 50, 411-426. SPENCE,A. M. (1974): Market Signaling. Cambridge: Harvard University Press.