Interim efficient allocations under uncertainty

This paper considers an exchange economy under uncertainty with asymmetric information. Uncertainty is represented by multiple priors and posteriors of agents who have either Bewley's incomplete preferences or Gilboa-Schmeidler's maximin expected utility preferences. The main results characterize interim efficient allocations under uncertainty; that is, they provide conditions on the sets of posteriors, thus implicitly on the way how agents update the sets of priors, for non-existence of a trade which makes all agents better off at any realization of private information. For agents with the incomplete preferences, the condition is necessary and sufficient, but for agents with the maximin expected utility preferences, the condition is sufficient only. A couple of necessary conditions for the latter case are provided.     

Perfect Bayesian implementation

Summary We study Social Choice Sets (SCS) implementable as perfect Bayesian equilibria of some incomplete information extensive form game. We provide a necessary condition which we callcondition . The condition is analogous tocondition C that Moore and Repullo [1988] show to be necessary for subgame perfect implementation in games of complete information, and it is weaker than the Bayesian Monotonicity condition stated in Jackson [1991]. Our first theorem establishes that Incentive Compatibility, Closure and Condition are necessary for implementation.Our second theorem establishes sufficient conditions. We show that any SCS which satisfies Incentive Compatibility, Closure and a condition called Sequential Monotonicity No Veto (SMNV) is implementable. SMNV is similar in spirit but weaker than the Monotonicity No Veto condition stated in Jackson [1991]. It is also similar to a combination of condition and No Veto Power, which Abreau and Sen show to be sufficient for implementation in subgame perfect equilibrium.This paper is a revised version of a chapter of my dissertation at Stanford University, Graduate School of Business. I would like to thank, without implicating, Faruk Gul, Bob Wilson and especially John Roberts for their constant advice and encouragement. I would also like to thank participants at the 1993 Summer Meeting of the Econometric Society and a referee for comments leading to substantial improvement in the paper. Financial support from Bocconi University is gratefully acknowledged.     

Robust implementation in general mechanisms

A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation.Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.     

Bayesian–Walrasian equilibria: beyond the rational expectations equilibrium

In general rational expectations equilibrium (REE), as introduced in Radner (Econometrica 47:655–678, 1978) in an Arrow–Debreu–McKenzie setting with uncertainty, does not exist. Moreover, it fails to be fully Pareto optimal and incentive compatible and is also not implementable as a perfect Bayesian equilibrium of an extensive form game (Glycopantis et al. in Econ Theory 26:765–791, 2005). The lack of all the above properties is mainly due to the fact that the agents are supposed to predict the equilibrium market clearing price (as agent’s expected maximized utility is conditioned on the information that equilibrium prices reveal), which leads inevitably to the presumption that agents know all the primitives in the economy, i.e., random initial endowments, random utility functions and private information sets. To get around this problematic equilibrium notion, we introduce a new concept called Bayesian–Walrasian equilibrium (BWE) which has Bayesian features. In particular, agents try to predict the market-clearing prices using Bayesian updating and evaluate their consumption in terms of Bayesian price estimates, which are different for each individual. In this framework agents maximize expected utility conditioned on their own private information about the state of nature, subject to a Bayesian estimated budget constraint. Market clearing is not an intrinsic part of the definition of BWE. However, both in the case of perfect foresight and in the case of symmetric information BWE leads to a statewise market clearing; it then becomes an ex post Walrasian equilibrium allocation. This new BWE exists under standard assumptions, in contrast to the REE. In particular, we show that our new BWE exists in the well-known example in Kreps (J Econ Theory 14:32–43, 1977), where REE fails to exist. This work was done in the Spring of 2005, when EJB was a visiting professor at the University of Illinois.     

Undominated Nash Implementation with Collusion and Renegotiation

This article looks at implementation in economic environments when agents have perfect information about the state of the world, but cannot commit not to renegotiate bad outcomes or to collude against each other. If renegotiation satisfies a weak condition of disagreement point monotonicity, then any Pareto-efficient social choice function can be implemented if there are at least three agents who play undominated Nash equilibria. The mechanism does not use modulo or integer games, has no bad mixed strategy equilibria, and is “bounded.”Journal of Economic LiteratureClassification Number: D71.     

Implementation in Economic Environments with Incomplete Information: The Use of Multi-Stage Games

This paper shows that, in economic environments with incomplete information, incentive compatibility and a preference reversal condition are sufficient for implementation in sequential equilibrium.Journal of Economic LiteratureClassification Numbers: C72, D71, D82.     

Equilibrium without prices: A central issue in pareto’smanuale

Pareto in hisManuale asserts that the price concept is inessential for the notion of equilibrium. This view deeply contrasts with the Arrow-Debreu’s dominant approach of Walrasian derivation. This identifies general equilibrium with a system of prices at which the decentralized choices of economic agents match. In this paper the author justifies and gives substance to the Paretian point of view. The differential merit of the Paretian approach is its independence of assumptions of price taking and of convexity in preferences and in production possibility sets. These assumptions are necessary, in the walrasian framework, only to maintain the centrality of prices, but they clash against the request of realism. (JEL:BOO, D51, D60)     

Mechanism Design under Alternative Information Structures and Constrained Capacity

In applied models, the choice of a particular incomplete information structure appears to have been motivated primarily by technical convenience. The information structures used can be classified as either probabilistic or partitional. Information is probabilistic if no agent can rule out any type profile of the remaining agents and, for at least one type of one agent, the conditional and marginal probability distributions over the remaining agents' types are not equal. Information is partitional if the only information the agents have is that one or more agents (individually) can rule out type profiles of the remaining agents and, for at least one type of one agent, that agent has information about the remaining agents. Partitional information includes complete information as a special case. Existing results on complete information environments suggest that partitional information might simplify implementation problems. Within the context of an applied agency model in which capacity is constrained, we provide results that seem to challenge this intuition.Journal of Economic LiteratureClassification Numbers: C72, D82.     

An extensive form interpretation of the private core

Summary. The private core of an economy with differential information, (Yannelis (1991)), is the set of all state-wise feasible and private information measurable allocations which cannot be dominated, in terms of ex ante expected utility functions, by state-wise feasible and private information measurable net trades of any coalition. It is coalitionally Bayesian incentive compatible and also takes into account the information superiority of an individual. We provide a noncooperative extensive form interpretation of the private core for three person games. We construct game trees which indicate the sequence of decisions and the information sets, and explain the rules for calculating ex ante expected payoffs. In the spirit of the Nash programme, the private core is thus shown to be supported by the perfect Bayesian equilibrium of a noncooperative game. The discussion contributes not only to the development of ideas but also to the understanding of the dynamics of how coalitionally incentive compatible contracts can be realized. Received: July 21, 2000; revised version: January 16, 2001     

Implementation with partial provability

We extend implementation theory by allowing the social choice function to depend on more than just the preferences of the agents and allowing agents to support their statements with hard evidence. We show that a simple condition on evidence is necessary for the implementation of a social choice function f when the preferences of the agents are state independent and sufficient for implementation for any preferences (including state dependent) with at least three agents if the social planner can perform small monetary transfers beyond those called for by f. If transfers can be large, f can be implemented in a game with perfect information when there are at least two players under a boundedness assumption. For both results, transfers only occur out of equilibrium. The use of evidence enables implementation which is robust in the sense that the planner needs little information about agents? preferences or beliefs and agents need little information about each others? preferences. Our results are robust to evidence forgery at any strictly positive cost.     

On extensive form implementation of contracts in differential information economies

Summary. In the context of differential information economies, with and without free disposal, we consider the concepts of Radner equilibrium, rational expectations equilibrium, private core, weak fine core and weak fine value. We look into the possible implementation of these concepts as perfect Bayesian or sequential equilibria of noncooperative dynamic formulations. We construct relevant game trees which indicate the sequence of decisions and the information sets, and explain the rules for calculating ex ante expected payoffs. The possibility of implementing an allocation is related to whether or not it is incentive compatible. Implementation through an exogenous third party or an endogenous intermediary is also considered. Received: November 19, 2001; revised version: April 17, 2002 RID="*" ID="*" This paper comes out of a visit by Nicholas Yannelis to City University, London, in December 2000. We are grateful to Dr A. Hadjiprocopis for his invaluable help with the implementation of Latex in a Unix environment. We also thank Leon Koutsougeras and a referee for several, helpful comments. Correspondence to: N.C. Yannelis     

Finite supermodular design with interdependent valuations

This paper studies supermodular mechanism design in environments with arbitrary (finite) type spaces and interdependent valuations. In these environments, the designer may have to use Bayesian equilibrium as a solution concept, because ex-post implementation may not be possible. We propose direct (Bayesian) mechanisms that are robust to certain forms of bounded rationality while controlling for equilibrium multiplicity. In quasi-linear environments with informational and allocative externalities, we show that any Bayesian mechanism that implements a social choice function can be converted into a supermodular mechanism that also implements the original decision rule. The proposed supermodular mechanism can be chosen in a way that minimizes the size of the equilibrium set, and we provide two sets of sufficient conditions to this effect. This is followed by conditions for supermodular implementation in unique equilibrium.     

Unique Implementation of the Full Surplus Extraction Outcome in Auctions with Correlated Types

In auctions with correlated types it is possible to design mechanisms such that full surplus extraction can be obtained as the outcome of an equilibrium in which agents use (weakly) dominant strategies. However, it is not assured that the outcome is unique. We present an example in which no mechanism can yield the full surplus extraction outcome as the unique Bayesian equilibrium outcome. Next we show that in the standard auction model the multiplicity problem can be fully resolved using sequential mechanisms, i.e., we show that it is possible to obtain the full surplus extraction outcome as the unique perfect Bayesian equilibrium outcome.Journal of Economic LiteratureClassification Numbers: D44; D70.     

A note on the one-deviation property in extensive form games

In an extensive form game, an assessment is said to satisfy the one-deviation property if for all possible payoffs at the terminal nodes the following holds: if a player at each of his information sets cannot improve upon his expected payoff by deviating unilaterally at this information set only, he cannot do so by deviating at any arbitrary collection of information sets. Hendon et al. (1996. Games Econom. Behav. 12, 274–282) have shown that pre-consistency of assessments implies the one-deviation property. In this note, it is shown that an appropriate weakening of pre-consistency, termed updating consistency, is both a sufficient and necessary condition for the one-deviation property. The result is extended to the context of rationalizability.     

Ex post implementation

We analyze the problem of fully implementing a social choice set in ex post equilibrium. We identify an ex post monotonicity condition that is necessary and—in economic environments—sufficient for full implementation in ex post equilibrium. We also identify an ex post monotonicity no veto condition that is sufficient. Ex post monotonicity is satisfied in all single crossing environments with strict ex post incentive constraints.We show by means of two classic examples that ex post monotonicity does not imply nor is it implied by Maskin monotonicity. The single unit auction with interdependent valuations is shown to satisfy ex post monotonicity but not Maskin monotonicity. We further describe a Pareto correspondence that fails ex post monotonicity but satisfies Maskin monotonicity.     

Cardinal Bayesian allocation mechanisms without transfers

In allocating goods with no use of monetary transfers, random allocation mechanisms can be designed in order to elicit information on preference intensities. I study the nontransfer allocation of two ex-ante identical objects under Bayesian incentive compatibility, with symmetric agents and independent private valuations. I find the ex-ante utilitarian-optimal mechanism, in which the probability of receiving a specified object is used as “numeraire” to purchase probability units of the other object. I characterize this mechanism as an appropriate combination of lotteries, auctions and insurance. The latter element ensures that efficient auctions are feasible. If the problem is constrained to guarantee exactly one object per agent, then the optimal mechanism uses no information other than the agents? ordinal preferences.     

Subgame perfect implementation: A necessary and almost sufficient condition

We present a necessary and almost sufficient condition for subgame perfect implementation of social choice correspondences. In societies with at least three individuals, any social choice correspondence which satisfies no veto power and our necessary Condition α is subgame perfect implementable. Thus Condition α is analogous to monotonicity which, by Maskin's celebrated characterization, is necessary and, in a similar way, almost sufficient for Nash implementation.     

Nash implementing non-monotonic social choice rules by awards

Summary. By a slight generalization of the definition of implementation (called implementation by awards), Maskin monotonicity is no more needed for Nash implementation. In fact, a weaker condition, to which we refer as almost monotonicity is both necessary and sufficient for social choice correspondences to be Nash implementable by awards. Hence our framework paves the way to the Nash implementation of social choice rules which otherwise fail to be Nash implementable. In particular, the Pareto social choice rule, the majority rule and the strong core are almost monotonic (hence Nash implementable by awards) while they are not Maskin monotonic (hence fail to be Nash implementable in the standard framework).Received: 12 April 2004, Revised: 18 March 2005, JEL Classification Numbers: D 79.I thank François Maniquet, Efe Ok and Ipek Ozkal-Sanver for their valuable contributions. The paper is part of a project on Electoral System Design supported by Bilgi University Research Fund.     

A characterization of virtual Bayesian implementation

We provide a characterization of virtual Bayesian implementation in pure strategies for environments satisfying no-total-indifference. A social choice function in such environments is virtually Bayesian implementable if and only if it satisfies incentive compatibility and a condition we term virtual monotonicity. The latter is weaker than Bayesian monotonicity—known to be necessary for Bayesian implementation. Virtual monotonicity is weak in the sense that it is generically satisfied in environments with at least three alternatives. This implies that in most environments virtual Bayesian implementation is as successful as it can be (incentive compatibility is the only condition needed).     

Non-implementation of rational expectations as a perfect Bayesian equilibrium

Summary. We point out several conceptual difficulties of the rational expectations equilibrium concept. In particular we show that such an equilibrium need not be incentive compatible and need not be implementable as a perfect Bayesian equilibrium . A comparison of rational expectations equilibria with the private core is also provided. We conclude that the private core is a more appropriate concept to capture the idea of contracts under asymmetric information.Received: 15 December 2003, Revised: 18 November 2004, JEL Classification Numbers: C71, C72, D5, D82. Correspondence to: Nicholas C. YannelisWe wish to thank Dr A. Hadjiprocopis for his invaluable help with the implementation of Latex in a Unix environment.