Nash-implementation of competitive equilibria via a bounded mechanism

We deal with the problem of providing incentives for the implementation of competitive outcomes in a pure-exchange economy with finitely many households. We construct a feasible price-quantity mechanism, which fully implements Walras equilibria via Nash equilibria in fairly general environments. Traders’ preferences need neither to be ordered nor continuous. In addition, the mechanism is such that no pure strategy is weakly dominated, hence is bounded (in the sense of Jackson 1992). In particular it makes no use of any integer game.     

Equilibrium theory with a measure space of possibly satiated consumers

Walras equilibria may not exist when consumers’ preferences are possibly satiated. To overcome this difficulty, several extended notions of equilibria have been proposed and all reduce to Walras equilibria under nonsatiation and free disposal. This includes the notions of equilibria with slack (also called dividend equilibria) as by Drèze and Müller [J. Economic Theory 23 (1980) 131], Makarov [J. Mathematical Economics 8 (1981) 87], Aumann and Drèze [Econometrica 54 (1986) 1271], Mas-Colell [Equilibrium theory with possibly satiated preferences, in: Majumdar, M. (Ed.), Proceedings of the Essays in Honour of David Gale on Equilibrium and Dynamics, Macmillan, London, pp. 201–213], monetary equilibria as by Kajii [J. Mathematical Economics 25 (1996) 75], or weak equilibria as by Polemarchakis and Siconolfi [J. Mathematical Economics 22 (1993) 85], which are all defined when there are finitely many consumers. This includes also the notion of free disposal equilibrium, when markets clear in a weak sense, allowing free disposal. Our paper considers an economy with a measure space of consumers and provides a general existence result of equilibria for the various existing notions. This result extends in particular the result by Hildenbrand [Econometrica 38 (1970) 608] on the existence of Walras equilibria.     

Mechanism design for a solution to the tragedy of commons

We consider the problem of the tragedy of commons in cooperative production economies, and propose a mechanism to resolve this tragedy, taking into account that the coordinator cannot perfectly monitor each agent’s labor skill and each agent may have an incentive to overstate as well as understate his own skill. Even in such a situation, the mechanism implements the proportional solution (Roemer in Soc Philos Policy 6:74–92, 1989 and Roemer and Silvestre in J Econ Theory 59:426–444, 1993) in Nash and strong equilibria when it is played as a normal form game. Moreover, the mechanism triply implements the solution in Nash, subgame-perfect, and strong equilibria when it is played as a two-stage extensive form game. We are greatly thankful to Semi Koray, William Thomson, and an anonymous referee of this journal for their concrete and helpful comments on improving the paper. An earlier version of this paper was presented at the annual meeting of the Japanese Economic Association held at Hitotsubashi University in October 2001 and at the Conference on Economic Design held at NYU in July 2002. We are grateful to Takehiko Yamato for his useful comments in the former conference. We are also thankful to Tatsuyoshi Saijo, Kotaro Suzumura, and Yoshikatsu Tatamitani for their kind comments.     

Hierarchic competitive equilibria

Without an interiority or strong survival assumption, an equilibrium may not exist in the standard Arrow–Debreu model. We propose a generalized concept of competitive equilibrium, called hierarchic equilibrium. Instead of using standard prices we use hierarchic prices. Existence will be shown without a strong survival assumption and without a non-satiation condition on the preferences. Under standard assumptions this reduces to the Walras equilibrium. Hierarchic equilibria are weakly Pareto optimal and any Pareto optimum can be decentralized without a border condition. We prove the existence of a Pareto optimal hierarchic equilibrium under additional assumptions. Later, we establish a core equivalence result.     

A mechanism implementing the proportional solution

We consider the problem of a commonly owned technology which transforms a single input into a single output. We are interested in implementing a social choice rule called theproportional solution. We introduce a mechanism which implements the proportional solution in Nash, strong (Nash) and undominated Nash equilibria. In the mechanism each agent announces only two numbers which can be interpreted as the total output and her share of the total input-output combination. This paper was originally titled "Doubly implementing the proportional solution." I would like to thank my advisor William Thomson for his detailed comments and suggestions. I would also like to thank Jeffrey Banks and Sung-Whee Shin for their comments. Two anonymous referees and an editor’s comments improved this paper substantially.     

Implementation of Pareto efficient allocations

This paper considers Nash implementation and double implementation of Pareto efficient allocations for production economies. We allow production sets and preferences are unknown to the planner. We present a well-behaved mechanism that fully implements Pareto efficient allocations in Nash equilibrium. The mechanism then is modified to fully doubly implement Pareto efficient allocations in Nash and strong Nash equilibria. The mechanisms constructed in the paper have many nice properties such as feasibility and continuity. In addition, they use finite-dimensional message spaces. Furthermore, the mechanism works not only for three or more agents, but also for two-agent economies.     

Constrained suboptimality when prices are non-competitive

The paper addresses the following question: how efficient is the market system in allocating resources if trade takes place at prices that are not competitive? Even though there are many partial answers to this question, an answer that stands comparison to the rigor by which the first and second welfare theorems are derived is lacking. We first prove a “Folk Theorem” on the generic suboptimality of equilibria at non-competitive prices. The more interesting problem is whether equilibria are constrained optimal, i.e. efficient relative to all allocations that are consistent with prices at which trade takes place. We discuss an optimality notion due to Bénassy, and argue that this notion admits no general conclusions. We then turn to the notion of p-optimality and give a necessary condition, called the separating property, for constrained optimality: each constrained household should be constrained in each constrained market. If the number of commodities is less than or equal to two, the case usually treated in the textbook, then this necessary condition is also sufficient. In that case equilibria are constrained optimal. When there are three or more commodities, two or more constrained households, and two or more constrained markets, this necessary condition is typically not sufficient and equilibria are generically constrained suboptimal.     

Finitely additive exchange economies

It is shown that the core and the set of Walras allocations of a non-atomic exchange economy are equal, if the set A of agents is either countable or a continuum, and even if all subsets of A are admitted as coalitions. The set of Walras allocations is shown to be not empty. These results are obtained by use of finitely additive measures defined on the algebra of all subsets of A.     

A general equilibrium analysis of strategic arbitrage

We analyze a general equilibrium framework with Cournot arbitrageurs and with price-taking investors who are subjected to restricted participation constraints. Restricted participation may leave some arbitrage opportunities unexploited by investors.We show existence of Cournot–Walras equilibria with an endogenous number of arbitrageurs. The number of arbitrageurs is endogenous since they have to sink entry costs in order to arbitrage across the relevant markets. We characterize equilibria and analyze the effects on equilibrium prices and quantities of increased competition among arbitrageurs due to lower entry costs.     

The Aubin private core of differential information economies

Abstract We establish characterizations of Radner equilibrium allocations via private core notions in the framework of differential information economies with a finite number of states of nature and a measure space of agents that may have atoms. The commodity space is an ordered separable Banach space whose positive cone admits an interior point. We introduce the notion of Aubin private core and prove that it provides a characterization of Radner equilibrium allocations. We show that the Aubin private core is equivalent to Edgeworth private equilibria and to privately non-dominated allocations of suitable associated economies. Mathematics Subject Classification (2000): 91B50, 91B44 Journal of Economic Literature Classification: D51, D82, D11     

Coalitional stability in the location problem with single-dipped preferences: An application of the minimax theorem

This study considers a situation in which agents choose the location of a public facility from a street according to a given mechanism. Agents have single-dipped preferences over a set of feasible locations. We analyze coalitional behavior for any given mechanism for this situation. We identify a necessary and sufficient condition for a mechanism to possess a strong Nash equilibrium by applying the minimax theorem of von Neumann and Morgenstern (1944). We introduce a class of core solutions and show that these solutions are characterized by strong Nash implementability. As a byproduct of these results, we propose a simple mechanism that implements any core solution in strong Nash equilibria.     

Efficiency and information aggregation in auctions with costly information

Consider an auction in which k identical objects are sold to n > k bidders who each have a value for one object which can have both private and common components to it. Private information concerning the common component of the object is not exogenously given, but rather endogenous and bidders face a cost to becoming informed. If the cost of information is not prohibitively high, then the equilibrium price in a uniform price auction will not aggregate private information, in contrast to the costless information case. Moreover, for a wide class of auctions if the cost of information is not prohibitively high then the objects can only be allocated in a weakly efficient sense, and then only if the equilibrium proportion of endogenously informed agents is vanishing as the economy grows. In spite of these results, it is shown that there is a mechanism for which there exist equilibria and for which (weak) efficiency is achieved as the economy grows in the face of endogenous information acquisition.Received: 6 August 2001, Accepted: 27 July 2002, JEL Classification: C72, D44, D82Financial support under NSF grant SBR-9507912 is gratefully acknowledged. I thank Valentina Bali for conversations out of which this project grew, and Larry Ausubel for a very helpful discussion of an early version of this paper at the 1998 summer meetings of the Econometric Society from which I have borrowed in structuring the exposition of the paper. I also thank Andy Postlewaite, Jeroen Swinkels, and an associate editor and anonymous referee for helpful comments and suggestions on earlier drafts.     

On the possibility of efficient bilateral trade

I study a sequential process in which different pairs of traders bargain over the terms of trade of an indivisible good. I consider bothone-sided andtwo-sided offers based bargaining at the stage-game level. The sequential process is modelled as an infinite stage-game of incomplete information and the paper studies the efficiency properties of its equilibria. It is shown: With one-sided offers, all equilibria are long-run ex post efficient; with two-sided offers, examples of equilibria are constructed with widely varying efficiency properties. This paper is based on Chapter 1 of my Ph.D thesis.     

Correlated equilibrium as a stable standard of behavior

We study the concept of correlated equilibrium within the framework of social situations (Greenberg 1990) and find that the unique optimistic stable standard of behavior (OSSB) of an appropriately defined correlated situation is nothing but the set of correlated equilibria. We generalize this situation in a natural way to accommodate coalitional deviations, and define the strong correlated situation. As the unique OSSB of this new situation, we derive a strong refinement of the set of correlated equilibria which we call strong correlated equilibrium. We analyse our concept and compare it with other existing notions using several examples. Received: 23 May 1996 / Accepted: 6 October 1997     

Second‐Price Auctions with Different Participation Costs

This paper studies equilibria of second‐price auctions in independent private value environments with different participation costs. Two types of equilibria are identified: monotonic equilibria in which a bidder with a lower participation cost results in a lower cutoff for submitting a bid, and nonmonotonic equilibria in which a lower participation cost results in a higher cutoff. We show that there always exists a monotonic equilibrium, and further, that the monotonic equilibrium is unique for either concave distribution functions or strictly convex distribution functions with nonincreasing reverse hazard rates. There exist nonmonotonic equilibria when the distribution functions are strictly convex and the difference of the participation costs is sufficiently small. We also provide comparative static analysis and study the limiting properties of equilibria when the difference in bidders’ participation costs approaches zero.     

A note on the decomposition (at a point) of aggregate excess demand on the Grassmannian

This paper analyzes the properties of aggregate excess demand functions for economies with an arbitrary finite set of N commodities where agents face trading restrictions of a general, abstract form: their budget set is defined by K-dimensional planes in ^N. It is shown that, if there are at least K agents in the economy, the only general property satisfied by the value of aggregate excess demand and its derivative, at any arbitrary point, is Walras Law. The result is established by considering an economy where agents' preferences are of a ‘generalized Leontief' type.     

An existence theorem for Cournot–Walras equilibria in a monopolistically competitive economy

We establish an existence theorem for Cournot–Walras equilibria in a monopolistically competitive economy. Instead of the traditional approach which depends on Kakutani’s fixed point theorem, we employ the theories of aggregative games and best reply potential games. We show that, if there exists a representative consumer, under some conditions on preferences and production technologies, the profit maximization game is a (pseudo) best reply potential game. Hence, the existence of the equilibria is proved independently of the well known convex-valued assumption on the best responses. Although our assumptions result in the additive separability on a utility function of a representative consumer, the existence of increasing returns and indivisible productions can be allowed. In our model, it is shown that the game played by firms exhibits strategic substitutes whether the products of firms are substitutes or complements, and this plays an important role for the existence of the equilibria.     

Existence of equilibria in economies with externalities and non-convexities in an infinite-dimensional commodity space

We prove an equilibrium existence theorem for economies with externalities, general types of non-convexities in the production sector, and infinitely many commodities. The consumption sets, the preferences of the consumers, and the production possibilities are represented by set-valued mappings to take into account the external effects. The firms set their prices according to general pricing rules which are supposed to have bounded losses and may depend upon the actions of the other economic agents. The commodity space is L_∞(M,M,μ), the space of all μ-essentially bounded M-measurable functions on M.As for our existence result, we consider the framework of Bewley (1972). However, there are four major problems in using this technique. To overcome two of these difficulties, we impose strong lower hemi-continuity assumptions upon the economies. The remaining problems are removed when the finite economies are large enough.Our model encompasses previous works on the existence of general equilibria when there are externalities and non-convexities but the commodity space is finite dimensional and those on general equilibria in non-convex economies with infinitely many commodities when no external effect is taken into account.     

Implementation by demand mechanisms

This paper introduces a class of implementation mechanisms for exchange economies, calleddemand mechanisms. Each agent “demands” a consumption bundle, and equilibria are situations where each agent receives the bundle he asked for. A simple necessary and sufficient condition for implementation by a demand mechanism is derived.