Informed-principal problems in environments with generalized private values

Mylovanov, Timofiy; Tröger, Thomas Article Informed-principal problems in environments with generalized private values Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Mylovanov, Timofiy; Tröger, Thomas (2012) : Informed-principal problems in environments with generalized private values, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 7, Iss. 3, pp. 465-488, https://doi.org/10.3982/TE787 This Version is available at: https://hdl.handle.net/10419/150177 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Theoretical Economics 7 (2012), 465–488 1555-7561/20120465 Informed-principal problems in environments with generalized private values Tymofiy Mylovanov Department of Economics, University of Pennsylvania Thomas Tröger Department of Economics, University of Mannheim We provide a solution to the problem of mechanism selection by a privately informed principal in generalized-private-value environments. In a broad class of these environments, the mechanism-selection game has a perfect-Bayesian equilibrium that has a strong neologism-proofness property. Equilibrium allocations that satisfy this property are characterized in terms of the players’ incentive and participation constraints, and can be computed using standard methods. Keywords. Informed principal, mechanism design, private values, strong unconstrained Pareto optimum. JEL classification. D82, D86. 1. Introduction In many applications of mechanism design, the principal has private information that is not directly payoff-relevant to the agents, but may influence her design: a seller’s opportunity cost influences the design of her sales mechanism, a supplier’s valuation influences her design of a collusive agreement, a speculator’s prior beliefs, in environments with heterogeneous priors, influence the design of the bet she offers, and the weight a regulator puts on consumer surplus influences the design of her regulation scheme. In this paper, we solve the problem of mechanism selection by an informed principal in such “generalized-private-value” environments where the agents’ payoff functions are independent of the principal’s type. Mechanism selection by an informed principal differs fundamentally from mechanism design by a principal with no private information. The latter can be formulated, due to the revelation principle, as a maximization problem of the principal’s payoff function subject to the agents’ incentive and participation constraints. If, however, the principal has private information, then upon observing a mechanism proposal, the agents Tymofiy Mylovanov: [email protected] Thomas Tröger: [email protected] We would like to thank the co-editor Jeff Ely and three anonymous referees for very useful comments that helped improve the paper. Financial Support by the German Science Foundation (DFG) through SFB/TR 15 “Governance and the Efficiency of Economic Systems” and by the National Science Foundation through Grant 0922365 is gratefully acknowledged. Copyright ©2012 Tymofiy Mylovanov and Thomas Tröger. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at http://econtheory.org. DOI: 10.3982/TE787 466 Mylovanov and Tröger Theoretical Economics 7 (2012) may update their beliefs about the principal’s type. Hence, the proposal of a mechanism by an informed principal must be viewed as a move in a game and the maximization approach is not applicable (Myerson 1983). We focus on the perfect-Bayesian equilibria of the mechanism-selection game that have a strong “neologism-proofness” property. The idea of neologism-proofness was introduced by Farrell (1993).1Given an equilibrium, a set of types Ais self-signaling if types in this set gain by inducing the belief that the type is in A. Farrell argues that a statement that a type belongs to a self-signaling set is credible and should be believed. Applying this idea to the mechanism-selection game, we require that any observed deviation from an equilibrium mechanism should be accompanied by a “credible” belief. We call a belief credible if none of the principal-types who would suffer from the deviation is believed to make it, and those types who already enjoy the highest feasible payoff are also believed to not make the deviation. Our concept of credibility differs from Farrell’s (1993) classical definition because we do not require that all the types who gain from the deviation are believed to make it, and because we require that none of the types who already enjoy the highest feasible payoff makes the deviation. An equilibrium such that no credible and profitable deviation exists is called strongly neologism-proof. Our main result is that a strongly neologism-proof equilibrium exists in a broad class of environments with generalized private values.2This is in stark contrast to standard signaling games, where neologism-proof equilibria often do not exist, and weaker equilibrium concepts, such as those based on the intuitive criterion (Cho and Kreps 1987), are considered (Riley 2001).3 We allow the outcome space to be any compact metric space. Any finite number of players and types is permitted, as is any continuous payoff function (no single-crossing or any other structural property is required). We assume, however, that private information is stochastically independent across players. Apart from that, we only make one technical assumption (“separability”): there exists an allocation such that all agents’ participation and incentive constraints are satisfied with strict inequality. Separability is typically easy to verify in a given environment. For environments with a single agent, we show that separability is satisfied in any environment in which the agent’s private information is payoff-relevant for herself. The strongly neologism-proof equilibrium allocations are attractive because they do not rely on implausible out-of-equilibrium beliefs about the principal. In addition, the strongly neologism-proof equilibrium allocations can be characterized in terms of the players’ participation and incentive constraints. This is important because it implies that these allocations can be found via standard mechanism-design methods. By contrast, the existing literature provides no characterization of perfect-Bayesian equilibria of the informed-principal game in general private-value environments. 1Ideas similar to Farrell’s (1993)wereusedbyGrossman and Perry (1986) to motivate their concept of perfect sequential equilibrium. 2In an environment with verifiable types, de Clippel and Minelli (2004) show the existence of an equilibrium that satisfies a related notion of neologism-proofness. 3See also Mailath et al. (1993), who argue in favor of another solution concept—undefeated equilibrium—that is also weaker than strong neologism-proofness. Theoretical Economics 7 (2012) Informed-principal problems 467 Myerson (1983)andMaskin and Tirole (1990) were the first to consider the problem of mechanism selection by an informed principal. Myerson’s elegant analysis is based on an axiomatic approach and applies to environments with a finite outcome space. Maskin and Tirole consider a class of single-agent environments with two possible types of the agent under particular structural assumptions about the outcome space and the players’ payoff functions.4Part of our contribution can be seen as extending ideas of Maskin and Tirole to a much more general setting. Maskin and Tirole’s analysis evolves around the concept of a “strong unconstrained Pareto optimum” (SUPO), which characterizes the entire set of perfect-Bayesian equilibrium allocations under some assumptions. Maskin and Tirole (cf. 1990, footnote 23 and, also, footnote 7in this paper) state that SUPO can be recast in terms of Farrell’s neologism-proofness. In our more general setting, it is more convenient to put a neologism-proofness property in the center right away, rather than trying to generalize the concept of a SUPO. Also, a straight generalization of SUPO would lead to existence problems. Our approach yields the existence of a strongly neologism-proof equilibrium even when no SUPO exists (cf. footnote 9). The basic reason why a privately informed principal’s mechanism may differ from the mechanism that she would offer if her information were public is simple. A privately informed principal may propose a mechanism that is independent of her private information, while she herself is a player in her mechanism. In such a mechanism, the agents’ incentive and participation constraints must hold only on average over the principal’s types, given the agents’ belief. The principal may be able to gain from such a weakening of the constraints. The crucial implication of the assumption of generalized private values is that the form of the agents’ incentive and participation constraints is independent of the principal’s type. As observed by Maskin and Tirole (1990), this makes it possible to interpret the different types of the principal as traders in a fictitious economy where each constraint corresponds to a good. The principal-types trade amounts of slack allowed for the various constraints. Any competitive equilibrium in this fictitious economy corresponds to an allocation that is a strongly neologism-proof equilibrium allocation of the mechanism-selection game. Technically, our main contribution is the result that a competitive equilibrium exists for the fictitious economy. To guarantee existence, we include the possibility that some goods have the price 0 and we allow free disposal (that is, in equilibrium, any constraint may be satisfied with strict inequality on average over the principal’s types). A further complication arises from the fact that the traders’ “utility functions” in the fictitious economy are determined endogenously in a way that their continuity cannot be guaranteed. Finally, Walras’ law may fail. Consequently, while our existence proof builds on Debreu’s (1959) classical arguments, some details are substantially different. By comparison, in Maskin and Tirole’s setting, it is sufficient to ignore trade in all but two constraints that have positive prices, traders’ utility functions in the fictitious economy are differentiable, and Walras’ law holds, so that Debreu’s arguments extend straightforwardly. 4Quesada (2010) provides conditions for equilibrium allocations in Maskin and Tirole (1990) to be deterministic and shows that their characterization continues to hold in a less restrictive environment. 468 Mylovanov and Tröger Theoretical Economics 7 (2012) The main result in our paper is a general existence result. By contrast, Maskin and Tirole (1990), Fleckinger (2007), Cella (2008), and Skreta (2011) focus on whether the privacy of the principal’s information allows the principal to improve her payoff. (For examples of an environment with private values in which the principal can benefit from the privacy of her information, see Section 4.) Severinov (2008) obtains the full-surplus extraction result for environments with the informed principal and correlated types. A few papers consider standard private values environments with continuous type spaces and quasilinear preferences. Yilankaya (1999) considers a standard bilateraltrade environment à la Myerson and Satterthwaite (1983), with the seller being the principal. It is shown that in this environment, the privacy of the principal’s information does not matter. A similar result is obtained by Tan (1996) for a procurement setting with multiple agents, and by Mylovanov and Tröger (2008) for extensions of Myerson’s (1981) optimal-auction environments and for the quasilinear versions of the principal– agent environments of Guesnerie and Laffont (1984) in which the principal is privately informed. In this paper, we analyze environments with generalized private values. We do not know how to extend our approach to the environments with common or interdependent values. The difficulty is that the market clearing condition (4) in the definition of the competitive equilibrium is not independent from the allocation of the slack consumption among different types of the principal if the agents’ payoffs depend on the principal’s type. Informed-principal problems in common- and interdependent-value environments are considered in Myerson (1983), Maskin and Tirole (1992), Severinov (2008), Skreta (2011), and Balkenborg and Makris (2010). Maskin and Tirole (1992, Proposition 7) provide necessary and sufficient conditions for the existence of neologism-proof equilibria under the assumption that the agent has no private information. Finally, there exists a separate literature that studies the informed-principal problem in moral-hazard environments, rather than in adverse-selection environments considered here (see, for example, Beaudry 1994,Chade and Silvers 2002,andKaya 2010). The rest of the paper is organized as follows. Section 2 describes the model. In Section 3, we characterize strong neologism-proofness in terms of incentive and participation constraints. In Section 4,wegiveexamples. Section 5 deals with the existence of strongly neologism-proof equilibria. Section 6 relates to other solution concepts. Section 7 concludes. Some proofs are given in the Appendix. 2. Model We consider the interaction of a principal (player 0) and nagents (players i∈N= {1n}). The players must collectively choose an outcome from a compact metric space of basic outcomes Z.5Every player i=0nhas a type tithat belongs to a finite type space Ti.Atype profile is any t∈T=T0×···×Tn. Sometimes we use the notation 5Hence, in environments with monetary transfers, the set of feasible transfers is truncated at some (arbitrarily high) point. Theoretical Economics 7 (2012) Informed-principal problems 469 T=Ti×T−i,t=(tit−i),ort=(t0tit−0i).Playeri’s payoff function, ui:Z×T→R is assumed to be continuous (note that the continuity assumption is void if Zis finite). We focus on environments that are characterized by the property that the agents’ payoff functions are independent of the principal’s type. Definition 1. An environment features generalized private values if, for all i=0, ui(z (t0t−0)) =ui(z (t 0t−0)) for all zt0t 0t−0 An outcome is a probability measure over basic outcomes; let Zdenote the set of outcomes. We identify any z∈Zwith the point distribution that puts probability 1 on the point z;hence,Z⊆Z. We extend the definition of uito Z×Tvia the statistical expectation. Some outcome z0∈Zis designated as the disagreement outcome. Every player’s payoff from the disagreement outcome is normalized to 0 (for every profile of other players’ types). The interaction is described by the following mechanism-selection game.First,for each player i, nature chooses a type. Let pi(ti)>0denote the probability of type ti∈Ti. We assume that types are stochastically independent. Each player privately observes her type ti. Second, the principal offers a mechanism M, which is a finite perfect-recall game form with players N∪{0}and with outcomes in Z. Third, the agents decide simultaneously whether to accept M.IfMis accepted unanimously, then each player chooses a message (consisting of an action at each of her information sets) in Mand the outcome specified by Mis implemented. If at least one agent rejects M, then the disagreement outcome z0is implemented.6 An allocation is a function ρ:T→Z that assigns an outcome ρ(t)to every type profile t∈T. Thus, an allocation describes the outcome of the mechanism-selection game as a function of the type profile. Alternatively, an allocation ρcan be interpreted as a direct mechanism, where the players i= 0nsimultaneously announce types ˆ ti(=messages) and the outcome ρ(ˆ t0ˆ tn)is implemented. The definition of the mechanism-selection game includes the possibility of proposing indirect mechanisms as well. 6Myerson (1983) and Maskin and Tirole (1990,1992) define similar mechanism-selection games. Both earlier models, however, restrict attention to simultaneous-move mechanisms, and Maskin and Tirole assume that players can use a public randomization device to decide which equilibrium to play in M.Myerson assumes that an agent’s decision to accept or reject is taken simultaneously with the choice of a message in M(other private actions beyond acceptance and rejection may also be allowed). The equilibrium allocations that we find are also equilibrium allocations in the games of Myerson (1983) and Maskin and Tirole (1990,1992). By contrast, the results in this paper do not apply if private actions can be taken sequentially as occurs, for example, in a model in which a rejection of the principal’s proposal is followed by a play of a status quo mechanism. 470 Mylovanov and Tröger Theoretical Economics 7 (2012) Aperfect-Bayesian equilibrium for the mechanism-selection game specifies (i) for each type of the principal, an optimal (possibly randomized) mechanism proposal, (ii) for each mechanism, a belief about the principal’s type that is computed via Bayes rule if the mechanism is proposed by at least one type, and (iii) for each mechanism, a strategy profile that is a sequential equilibrium in the continuation game that follows the proposal of the mechanism. A well known drawback of the perfect-Bayesian equilibrium concept is that the belief about the principal’s type remains unrestricted if the principal proposes an “off-path” mechanism that no type was expected to propose. This may, in principle, allow for rather implausible equilibria. Hence, rather than attempting to find all equilibria, we focus on the existence and characterization of equilibria that have an additional property called strong neologism-proofness.7A similar approach was pioneered by Farrell (1993)inthe context of standard signaling games (related ideas were put forward by Grossman and Perry 1986). 3. Strongly neologism-proof allocations In this section, we define strongly neologism-proof allocations and show that any such allocation is a perfect-Bayesian equilibrium outcome. Hence, strong neologismproofness is a refinement of perfect-Bayesian equilibrium. Additional notation is required. Consider the continuation game that begins after some arbitrary mechanism is proposed. Let q0:T0→[01]denote the probability distribution that describes the agents’ belief about the principal’s type at the beginning of the continuation game. Let ρdenote the allocation resulting from the acceptance or rejection decisions and the subsequent play of the mechanism. The expected payoff of type tiof player iif she follows the rejection or acceptance and message choice of type ˆ tiis Uρq0 i(ˆ titi)= t−i∈T−i ui(ρ(ˆ tit−i)(tit−i))q−i(t−i) where q−i(t−i)=q0(t0)·p1(t1)···pi−1(ti−1)·pi+1(ti+1)···pn(tn)if i= 0and q−0(t−0)= p1(t1)·····pn(tn). The expected payoff of type tiof player ifrom allocation ρis Uρq0 i(ti)=Uρq0 i(titi) We use the shortcut Uρ 0(t0)=Uρq0 0(t0), which is justified by the fact that the principal’s expected payoff is independent of q0. Definition 2. An allocation ρis called q0-feasible if, given the belief q0and using the direct-mechanism interpretation of ρ, no type of any player has an incentive to reject ρ 7Extrapolating a result of Maskin and Tirole (1990, Proposition 7) and further examples, one may conjecture that all perfect-Bayesian equilibria are strongly neologism-proof, but showing this generally appears to be beyond reach. Theoretical Economics 7 (2012) Informed-principal problems 471 or to deviate from announcing her true type: for all i, Uρq0 i(ti)≥Uρq0 i(ˆ titi)for all tiˆ ti(1) Uρq0 i(ti)≥0for all ti(2) By the revelation principle, any perfect-Bayesian equilibrium allocation of the mechanism-selection game is p0-feasible. Hence, as observed by Myerson (1983), without loss of generality, we may restrict attention to perfect-Bayesian equilibria in which all types of the principal offer the same p0-feasible allocation as a direct mechanism (“principle of inscrutability”). However, as far as continuation equilibria following offpath mechanism proposals are concerned, we cannot restrict attention to p0-feasible allocations, but have to consider q0-feasible allocations for arbitrary beliefs q0. In general, off-path beliefs q0have to be different from the prior belief p0so as to make deviating mechanisms unattractive for all types of the principal.8 Given any allocations ρand ρ, the set of principal-types that are strictly better off in ρis denoted Sρρ={t0∈T0|Uρ 0(t0)>Uρ 0(t0)} The set of types who in ρobtain the highest feasible payoff is denoted H(ρ) =t0∈T0Uρ 0(t0)= t−0∈T−0 max z∈Zu0(z t0t−0)q(t−0) Given any allocations ρand ρ, we say that a belief q0about the principal’s type is credible for ρrelative to ρif it is consistent with Bayesian updating given the following behavior: none of the principal-types who is strictly better off in ρthan in ρor who already enjoys the highest feasible payoff in ρ, chooses ρ,thatis, ∀t0∈S(ρρ)∪H(ρ):q0(t0)=0 Let Supp(q0)denote the support of q0. Definition 3. An allocation ρis called strongly neologism-proof if ρis p0-feasible and there exists no belief q0together with a q0-feasible allocation ρsuch that (i) q0is credible for ρrelative to ρand (ii) S(ρρ)∩Supp(q0)=∅. Note that the credibility of a belief q0does not reflect any requirement that the types who are strictly better off in ρthan in ρwould actually choose ρ.Thisaspectofour concept of credibility is more general than Farrell’s original definition; it reflects that 8An instructive example is provided by Yilankaya (1999). He considers a standard bilateral-trade environment à la Myerson and Satterthwaite (1983), with the seller being the principal. A perfect-Bayesian equilibrium allocation is constructed from optimal fixed-price offers by all types of the seller. Some types would gain by deviating to a double-auction mechanism if the buyer kept her prior belief. If, however, the buyer believes that the lowest cost seller proposes the double auction, this deviation becomes unprofitable for all types of the seller. 472 Mylovanov and Tröger Theoretical Economics 7 (2012) we do not want to exclude the possibility that some types, although preferring ρover ρ, may not be believed to be among the deviators (possibly because there may exist another deviation that is more attractive than ρ). A second novel aspect of our concept of credibility is our requirement that no type who already enjoys the highest feasible payoff in ρmay be attracted to ρ. This restriction is necessary for our general existence result (see Example 1 and the proof of Lemma 3 below). But in many environments, including all environments where sufficiently large monetary transfers are feasible, the restriction is not binding: Remark 1. Suppose that the outcome space allows such large monetary transfers between the players that in any allocation that specifies the largest feasible transfer to the principal, at least one agent’s participation constraint is violated. Then H(ρ) =∅for any p0-feasible allocation ρ. This remark applies to most environments considered in the earlier literature. Hence, in all these environments, our concept of neologism-proofness is more demanding than Farrell’s (which strengthens our existence result). It remains to show that strongly neologism-proof allocations actually are perfect- Bayesian equilibrium allocations. Proposition 1. Any strongly neologism-proof allocation is a perfect-Bayesian equilibrium allocation of the mechanism-selection game. Proof. Consider any strongly neologism-proof allocation ρ. We construct a perfect- Bayesian equilibrium of the mechanism-selection game as follows. All types of the principal propose the direct mechanism ρ.Ifρis proposed, all agents accept and all players announce their true types. It remains to construct, for any mechanism M=ρ, the agents’ belief qMabout the principal’s type and the strategy profile τMfor the continuation game that begins when Mis proposed. Fix some M=ρand consider the following game G(M): First, nature chooses privately observed types t0tnexactly as in the mechanismselection game. Second, if t0∈H(ρ), then the game ends (we may specify arbitrary payoffs in this case). However, if t0/∈H(ρ), then the principal chooses between two actions. One action ends the game and she obtains the payoff Uρ 0(t0)(we may specify arbitrary payoffs for the agents in this case); the other action is to offer the mechanism M. Third, the agents decide simultaneously whether to accept M.IfMis accepted unanimously, then each player chooses a message in Mand the outcome specified by Mis implemented. If at least one agent rejects M, then the disagreement outcome z0is implemented. This is a finite game with perfect recall and thus has a sequential equilibrium σM.Define qMas any belief about the principal at the beginning of the third stage of the game G(M) that is consistent with σM.DefineτMas the strategy profile induced by σMin the continuation game that begins at the third stage of G(M). Theoretical Economics 7 (2012) Informed-principal problems 479 Proof. Suppose that γ∗·c<0for some maximizer cin (3). Let ρbe a maximizer of problem J(t0c).Then Uρ 0(t0)=V(t 0c)=V(t 0c∗ t0)=Uρ 0(t0) implying that t0/∈H(ρ). Hence, there exists a type profile t−0such that ρ(t)puts probability less than 1 on the outcomes in argmaxz∈Zu0(zt). Define an allocation ρ such that ρ(t)∈argmaxz∈Zu0(z t)and ρ(t)=ρ(t)for all type profiles t=t. Consider c:G→Rg→c(g) + with >0so small that γ∗·c<0(5) The allocation ρsatisfies all constraints of problem J(t0c)with strict inequality. Hence, an allocation ρ that implements ρwith probability λ<1and ρ with probability 1−λbelongs to the feasible region of problem J(t0c)if λis sufficiently close to 1, implying V(t 0c)≥Uρ 0(t0)>Uρ 0(t0)=V(t 0c) Moreover, by (5), the point csatisfies the constraint of the problem in (3). But this contradicts the assumption that cis a maximizer of the problem in (3).  We are now ready to connect slack-exchange equilibrium and strong neologismproofness. The result parallels the first welfare theorem for competitive equilibria. Lemma 3. Any slack-exchange equilibrium allocation in any environment with generalized private values is strongly neologism-proof. Proof.Letρbe a slack-exchange equilibrium allocation. To show that ρis p0-feasible, observe first that, by (4), ρsatisfies (1)and(2) for all i=0. Because the allocation that implements the disagreement outcome is feasible in problem J(t00),(2)issatisfiedfori=0. Finally, (1)issatisfiedfori=0because the bundle (r∗ ˆ t0c∗ ˆ t0)belongs to the feasible region of problem (3). Suppose that ρis not strongly neologism-proof. Then there exists a belief q0together with a q0-feasible allocation ρsuch that (i) q0is credible for ρrelative to ρand (ii) at least one principal-type t 0is better off in ρthan in ρ;thatis, Uρ 0(t0)≥Uρ 0(t0)for all t0∈Supp(q0)(6) and Uρ 0(t 0)>Uρ 0(t 0) t 0∈Supp(q0) (7) 480 Mylovanov and Tröger Theoretical Economics 7 (2012) For all t0∈Supp(q0),definec t0such that ρsatisfies all constraints of problem J(t0c t0) with equality. Because ρis q0-feasible,  t0∈T0 c t0(g)q0(t0)≤0for all g∈G(8) Using Lemma 2 together with (6), we find γ∗·c t0≥0for all t0∈Supp(q0) (9) Similarly, using (7), γ∗·c t 0 >0(10) Building a weighted sum from (9)and(10), we obtain  t0∈T0 g∈G γ∗(g)c t0(g)q0(t0)>0 which yields a contradiction to (8).  Existence of competitive equilibria in the fictitious economy The lemma below is the last step toward proving our main result, Proposition 2.Herewe use the separability assumption. It guarantees that the budget set of any trader in the slack-exchange economy has an interior point, which is crucial toward showing that her demand correspondence is upper hemicontinuous. Lemma 4. A slack-exchange equilibrium exists in any separable environment with generalized private values. Our basic line of proof is—like the proof of the corresponding result of Maskin and Tirole (1986,1990)—inspired by Debreu (1959). The main complication relative to Debreu and Maskin and Tirole arises from the fact that the utility function V(t 0·)of any trader t0is not exogenously given, but is endogenously derived as the solution value of a maximization problem. In particular, the continuity of the objective, which is required in Debreu’s arguments, is—in contrast to the situation in Maskin and Tirole’s model— not obviously satisfied (the nonobvious assumption of Berge’s Maximum Theorem is the lower-hemicontinuity of the feasible region of problem J(t0rc)). We circumvent the continuity proof, showing only that V(t 0·)is upper semicontinuous (15), which captures the absence of downward jumps (see, e.g., Luenberger 1969, p. 40). Hence, by a generalized version of Weierstrass’ theorem, an optimal bundle of slacks (i.e., a solution to the problem in (3)) always exists. We use the upper-semicontinuity together with the concavity of V(t 0·)and the existence of an interior point of the trader’s budget set to show that (16), the solution value of problem (3), is lower semicontinuous in the price vector γ. The lower-semicontinuity of the solution value implies the upperhemicontinuity of the demand correspondence, which is the core step toward applying Kakutani’s fixed-point theorem to show equilibrium existence. The complete proof can be found in the Appendix. Theoretical Economics 7 (2012) Informed-principal problems 481 6. Relation to other solution concepts In this section, we explain how strongly neologism-proof equilibrium is related to other solution concepts that have been proposed for informed-principal problems. The technical concept of a strong unconstrained Pareto optimum (SUPO) plays a major role in the analysis of Maskin and Tirole (1990). A p0-feasible allocation ρis a SUPO if there exists no other allocation ρtogether with a belief q0about the principal’s type such that the agents’ (not necessarily the principal’s!) incentive and participation constraints are satisfied for ρandsuchthatρis weakly preferred to ρby all types of the principal, and strictly so for at least one type and strictly so for all types if q0does not have full support. Maskin and Tirole (1990, footnote 23) observe that SUPO is equivalent to Farrell’s (1993) neologism-proofness, as adapted to their setting. Because in their setting, neologism-proofness is implied by strong neologism-proofness (cf. Remark 1), we can conclude from Maskin and Tirole’s observation that SUPO is implied by strong neologism-proofness in their setting. Myerson (1983) proposes neutral optimum as an axiomatically founded solution concept that always exists in environments with finite outcome spaces and finite type spaces. We do not know the exact relation between neutral optimum and strong neologism-proofness, even in environments with generalized private values. However, a clear relation to another solution concept introduced by Myerson (1983) can be established: strongly neologism-proof allocation ⇐ ⇒ core allocation An allocation ρis a core allocation if (i) ρis p0-feasible and (ii) there exists no allocation ρsuch that ρis q0-feasible for all beliefs q0such that S(ρρ)=∅and q0(t0)=p(t0) t 0∈Sp(t 0)(S(ρρ)⊆S⊆T0) Setting S=S(ρρ)in this definition, it follows that any strongly neologism-proof equilibrium allocation is a core allocation. Alternatively, in Example 2,anyp0-feasible allocation in which each type of the principal obtains at least the expected payoff 1 is a core allocation, showing that not all core allocations are strongly neologism-proof equilibrium allocations. Finally, observe that any strongly neologism-proof allocation satisfies the intuitive criterion (as adapted to our setting). To see this, consider an allocation ρthat violates the intuitive criterion. Then there exists a principal-type t0and a mechanism Msuch that, for any belief q0that is reasonable when Mis proposed, in any sequential equilibrium allocation ρof the continuation game when Mis proposed, the expected payoff of type t0is larger than her payoff in ρ. The belief q0that puts probability 1 on type t0 is reasonable. Hence, q0is credible for ρrelative to ρ, implying that ρis not strongly neologism-proof. 482 Mylovanov and Tröger Theoretical Economics 7 (2012) 7. Conclusion In this paper, we offer a solution to the informed principal problem in the environments with generalized private values. We demonstrate that there exists a perfect-Bayesian equilibrium that is strongly neologism-proof. The equilibrium outcomes can be characterized in terms of the agents’ incentive and participation constraints. This makes the problem more tractable. The proof relies on demonstrating the existence of a competitive equilibrium in a fictitious economy in which different types of the principal trade slacks in the agents’ incentive and participation constraints. Appendix Proof of “if”in Remark 2. In Step 1, we show that the set of agent types can be partitioned into subsets such that to each subset an outcome is assigned that all types in this subset strictly prefer to the disagreement outcome as well as to the outcomes assigned to the other subsets. In Step 2, we show that to each agent type an outcome can be assigned that the agent strictly prefers to the outcomes assigned to the other agents. We then define an allocation such that each agent type obtains a convex combination of the outcome assigned to her subset in Step 1 and the outcome assigned to her in Step 1. If in this convex combination, the Step 1 outcome has a sufficiently large weight, then the allocation satisfies the incentive and participation constraints with strict inequality for all types. To prove Step 1, one constructs a partition inductively. By (ii), there exists an outcome z1that is preferred to the disagreement outcome by at least one agent type, and let P1be the set of types that strictly prefer the outcome to the disagreement outcome. If P1=T1, then, by (ii), some other outcome zis strictly preferred to disagreement by a subset P2of the remaining types. Defining z2as a convex combination of the disagreement outcomes z0and z, and putting enough weight on z0, we can guarantee that the types in P1prefer z1to z2.ThetypesinP1have the reverse preference because they prefer z2to z0to z1. This construction can be continued until a partition is obtained. Step 2 is also proved inductively. Start with any two agent types. By (i), one can assign two outcomes over which the types have opposite strict preferences. Pick a third type. From the outcomes assigned to the first two types, identify one that is most preferred by the third type. Then one perturbs the outcome assignments such that preferences become strict. This can be done by weighting in the most or less preferred outcome with a small probability. This construction can be continued until outcomes are assigned to all types. Step 1. There exists a partition P1Pk(k≥1)ofT1and outcomes z1zk∈Z such that, for all j=1k, ∀t1∈Pj:u1(zjt1)>u 1(z0t1) ∀t1∈Pjl∈{1k}\{j}:u1(zjt1)>u 1(zlt1) To prove this, consider for any k=12, the following statement (∗k): Theoretical Economics 7 (2012) Informed-principal problems 483 There exist pairwise-disjoint nonempty sets P1Pk⊆T1and outcomes z1zk∈Z such that, for all j=1k, ∀t1∈Pj:u1(zjt1)>u 1(z0t1) ∀t1∈Pjl∈{1k}\{j}:u1(zjt1)>u 1(zlt1) ∀t1∈T1\(P1∪···∪Pk):u1(zjt1)≤u1(z0t1) Statement (∗1)istrue:letz1∈Zbe an outcome that is strictly preferred to z0by at least one agent-type, and denote by P1the set of types that strictly prefer z1to z0. Let ˆ kbe the largest number such that (∗ˆ k)istrue(observethat ˆ k<∞because (∗k) fails for all k>|T1|). It is sufficient to show that P1∪···∪Pˆ k=T1. Suppose the opposite. Then there exists an outcome z∈Zthat is strictly preferred to z0by at least one agent-type in T1\(P1∪···∪Pˆ k).Define Pˆ k+1={t1∈T1\(P1∪···∪Pˆ k)|u1(zt1)>u 1(z0t1)} Now we can define zˆ k+1=λz0+(1−λ)zwith λ<1sufficiently close to 1 such that statement (∗(ˆ k+1)) is true. This contradicts the maximality of ˆ k. Step 2. For every t1∈T1, there exists an outcome ζ(t1)∈Zsuch that ∀t1t 1∈T1t 1=t1:u1(ζ(t1)t1)>u 1(ζ(t 1)t1) To prove this, consider for all P⊆T1the following statement (∗P): For every t1∈P, there exists an outcome ζ(t1)∈Zsuch that ∀t1t 1∈Pt 1=t1:u1(ζ(t1)t1)>u 1(ζ(t 1)t1) If Pis a singleton, then (∗P) is clearly true. Let ˆ Pa set of maximal cardinality with the property that (∗ˆ P) is true. It is sufficient to show that ˆ P=T1. Suppose the opposite. Choose any s1∈T1\ˆ P.Define z∈argmax z∈Z u1(z s1) z ∈argmin z∈Z u1(z s1) Let S1=argmaxt1∈ˆ Pu1(ζ(t1) s1)and choose any s 1∈S1.Definew(t1)=ζ(t1)for all t1∈ ˆ P\S1. Case 1. u1(ζ(s 1)s1)>u 1(zs1).Thendefinew(s 1)=ζ(s 1).Forallt1∈S1\{s 1},define w(t1)=λζ(t1)+(1−λ)z with some λ<1.Withλchosen sufficiently close to 1, statement (∗∗ ˆ P) holds: Statement (∗ˆ P) holds with ζ(·)replaced by w(·).Moreover,u1(w(t1) s1)<u 1(w(s 1)s1)for all t1∈ˆ P\{s 1}. Case 2: u1(ζ(s 1)s1)<u 1(zs1).12 For all t1∈S1\{s 1},definew(t1)=ζ(t1).Define w(s 1)=λζ(s1)+(1−λ)zwith some λ<1.Withλchosen sufficiently close to 1, statement (∗∗ ˆ P) holds. 12One of Cases 1 or 2 always occurs, because otherwise u1(zs1)=u1(zs1), implying that type s1is indifferent between all outcomes, which is impossible because there exists an outcome that she strictly prefers to the disagreement outcome. 484 Mylovanov and Tröger Theoretical Economics 7 (2012) Because the types s1and s 1have nonidentical preferences, there exist outcomes yy∈Zsuch that u1(ys1)>u 1(ys1) u1(y s 1)<u 1(ys 1) Define w(s1)=λw(s1)+(1−λ)y and w(s 1)=λw(s1)+(1−λ)y.Definew(t1)=w(t1) for all t1∈ˆ P\{s1s 1}.Withλchosen sufficiently close to 1, statement (∗(ˆ P∪{s 1})) holds with ζ(·)replaced by w(·), a contradiction to the maximality of ˆ P, completing Step 2. Now let P1Pk(k≥1)andz1zkbe as in Step 1, and let the function ζ(·)be as in Step 2. For all t1∈T1,definej(t1)such that t1∈Pj. Then the allocation ρdefined by ρ(t0t1)=λzj(t1)+(1−λ)ζ(t1) implies that all incentive and participation constraints of all agent types are satisfied with strict inequality if λis chosen sufficiently close to 1.  Proof of Lemma 4. Any consumption bundle in C, as well as any price vector, belongs to the Euclidean space R|G|. We use standard operators in Euclidean spaces such as +, min,or≤, all of which are defined componentwise. First we show that the set Cis closed(11) To see this, consider any sequence cm→csuch that cm∈C. By assumption, the constraint set of J(t0cm)contains a point ρm. For all sufficiently large m, the point ρm belongs to the feasible region of problem J(t0c +1),where1denotes the vector that is identically equal to 1. Because the latter feasible region is compact, ρmhas a subsequence that converges to some point ρ. By continuity, ρbelongs to the feasible region of J(t0c).Hence,c∈C. This completes the proof of (11). Because Zis compact, there exists a lower bound for the size of the left-hand side of every constraint of J(t0c). Hence, there exists c∈Csuch that V(t 0c)=V(t 0min{cc})for all c∈C (12) Similarly, there exists c∈R|G|such that C⊆{c|c≥c}(13) By (11), (12), and (13), the set D=C∩{c|c≤c}is compact(14) Define the unit simplex =γ∈R|G|γ≥0 g∈G γ(g) =1 Theoretical Economics 7 (2012) Informed-principal problems 485 For every γ∈, consider the problem E(t0γ): max c∈DV(t 0c) subject to γ·c≤0 The objective V(t 0·)of problem E(t0γ)is upper semicontinuous: for any convergent sequence (cm)in D, Vt0lim mcm≥limsup m V(t 0cm) (15) To see (15), let c=lim cmand let (cml)be a subsequence such that V(t 0cml)converges. Let ρlbe a maximizer of problem J(t0cml)and let (ρlk)be a subsequence such that ρlk converges. Because limkcmlk=c, the limit ρ=limkρlkbelongs to the feasible region of problem J(t0c).Hence, V(t 0c)≥Uρ 0(t0)=lim kUρlk 0(t0)=lim kV(t 0cmlk)=lim lV(t 0cml) By (15)andbecause,by(14), the feasible region of problem E(t0γ)is compact, a maximizer to problem E(t0γ)exists (see, e.g., Luenberger 1969, p. 40); let e(t0γ)denote the set of maximizers. The correspondence e(t0·):→Dis convex-valued because V(t 0·)is concave. To show that e(t0·)is upper hemicontinuous, we begin by showing that for every sequence in , if γm→γ then liminf mvt0γm≥v(t0γ) (16) where v(t0x) denotes the value reached at the maximum of problem E(t0x) for any x∈. Let c∗∈e(t0γ).Ifc∗·γ<0, then c∗·γm<0if mis sufficiently large, hence c∗belongs to the feasible region of E(t0γm),whichshows(16). Now suppose that c∗·γ=0(17) Using the separability assumption, the set Dcontains a strictly negative point c−<0. For all large m,define αm=min1−c−·γm c∗·γm−c−·γm(18) The convex combination cm=αmc∗+(1−αm)c−∈D. By construction, cmbelongs to the feasible region of problem E(t0γm). Hence, using the concavity of V(t 0·), αmV(t 0c∗)+(1−αm)V (t0c−)≤V(t 0cm)≤v(t0γm) (19) As m→∞,wehaveαm→1by (17)and(18). Hence, (19)implies V(t 0c∗)≤liminf mv(t0γm) Because V(t 0c∗)=v(t0γ),weobtain(16). 486 Mylovanov and Tröger Theoretical Economics 7 (2012) To show that e(t0·)is upper hemicontinuous, suppose that γm→γ,cm∈e(t0γm), and cm→c.Then V(t 0c)≥lim inf mV(t 0cm)=liminf mv(t0γm)≥v(t0γ) where the first inequality follows from (15) and the second inequality follows from (16). Hence, c∈e(t0γ)because cbelongs to the feasible region of E(t0γ). Define a correspondence h:t0∈T0D→by letting h((ct0)t0∈T0)be the set of solutions to the problem R((ct0)t0∈T0): max γ∈ t0∈T0 p(t0)γ ·ct0 By Berge’s maximum theorem, his upper hemicontinuous. Moreover, his convexvalued. By Kakutani’s theorem, the correspondence  t0∈T0 D×→ t0∈T0 D× (xγ) → t0∈T0 e(t0γ)×h(x) has a fixed point ((c∗ t0)t0∈T0γ∗). To complete the proof that ((c∗ t0)t0∈T0γ∗)is a slack-exchange equilibrium, it remains to show (4). Suppose that (4) fails; i.e., there exists g∈Gsuch that  t0∈T0 p(t0)c∗ t0(g) > 0(20) Choose γ∈such that γ(g)=0for all g=g.Then(20)impliesthat  t0∈T0 p(t0)γ ·c∗ t0>0 This contradicts the fact that γ∗solves problem R((c∗ t0)t0∈T0), because, using the constraint of problem E(t0γ∗)for all t0,  t0∈T0 p(t0)γ∗·c∗ t0≤0 References Balkenborg, Dieter and Miltiadis Makris (2010), “An undominated mechanism for a class of informed principal problems with common values.” Unpublished paper. [468] Beaudry, Paul (1994), “Why an informed principal may leave rents to an agent.” International Economic Review, 35, 821–832. [468] Theoretical Economics 7 (2012) Informed-principal problems 487 Billingsley, Patrick (1999), Convergence of Probability Measures, second edition. Wiley, New York. [477] Cella, Michela (2008), “Informed principal with correlation.” Games and Economic Behavior, 64, 433–456. [468] Chade, Hector and Randy Silvers (2002), “Informed principal, moral hazard, and the value of a more informative technology.” Economics Letters, 74, 291–300. [468] Cho, In-Koo and David M. Kreps (1987), “Signaling games and stable equilibria.” Quarterly Journal of Economics, 102, 179–221. [466] Cramton, Peter, Robert Gibbons, and Paul Klemperer (1987), “Dissolving a partnership efficiently.” Econometrica, 55, 615–632. [474] de Clippel, Geoffroy and Enrico Minelli (2004), “Two-person bargaining with verifiable information.” Journal of Mathematical Economics, 40, 799–813. [466] Debreu, Gerard (1959), Theory of Value. Yale University Press, New Haven, Connecticut. [467,480] Farrell, Joseph (1993), “Meaning and credibility in cheap-talk games.” Games and Economic Behavior, 5, 514–531. [466,470,481] Fleckinger, Pierre (2007), “Informed principal and countervailing incentives.” Economics Letters, 94, 240–244. [468] Grossman, Sanford J. and Motty Perry (1986), “Perfect sequential equilibrium.” Journal of Economic Theory, 39, 97–119. [466,470] Guesnerie, Roger and Jean-Jacques 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. [468] Kaya, Ayça (2010), “When does it pay to get informed?” International Economic Review, 51, 533–551. [468] Luenberger, David G. (1969), Optimization by Vector Space Methods. Wiley, New York. [480,485] Mailath, George J., Masahiro Okuno-Fujiwara, and Andrew Postlewaite (1993), “Beliefbased refinements in signalling games.” Journal of Economic Theory, 60, 241–276. [466] Maskin, Eric and Jean Tirole (1986), “Principals with private information, I: Independent values.” Discussion Paper 1234, Department of Economics, Harvard University. [480] Maskin, Eric and Jean Tirole (1990), “The principal–agent relationship with an informed principal: The case of private values.” Econometrica, 58, 379–409. [467,468,469,470, 474,478,480,481] Maskin, Eric and Jean Tirole (1992), “The principal–agent relationship with an informed principal, II: Common values.” Econometrica, 60, 1–42. [468,469] 488 Mylovanov and Tröger Theoretical Economics 7 (2012) Myerson, Roger B. (1981), “Optimal auction design.” Mathematics of Operations Research, 6, 58–73. [468] Myerson, Roger B. (1983), “Mechanism design by an informed principal.” Econometrica, 51, 1767–1797. [466,467,468,469,471,481] Myerson, Roger B. and Mark A. Satterthwaite (1983), “Efficient mechanisms for bilateral trading.” Journal of Economic Theory, 29, 265–281. [468,471] Mylovanov, Tymofiy and Thomas Tröger (2008), “Optimal auction design and irrelevance of privacy of information.” Unpublished paper. [468] Quesada, Lucia (2010), “A comprehensive note on the informed principal with private values and independent types.” Unpublished paper. [467] Riley, John G. (2001), “Silver signals: Twenty-five years of screening and signaling.” Journal of Economic Literature, 39, 432–478. [466] Severinov, Sergei (2008), “An efficient solution to the informed principal problem.” Journal of Economic Theory, 141, 114–133. [468] Skreta, Vasiliki (2011), “On the informed seller problem: Optimal information disclosure.” Review of Economic Design, 15, 1–36. [468] Tan, Guofu (1996), “Optimal procurement mechanisms for an informed buyer.” Canadian Journal of Economics, 29, 699–716. [468] Yilankaya, Okan (1999), “A note on the seller’s optimal mechanism in bilateral trade with two-sided incomplete information.” Journal of Economic Theory, 87, 267–271. [468,471] Submitted 2010-5-8. Final version accepted 2011-8-1. Available online 2011-8-2.