Walrasian analysis via two-player games

We associate to any pure exchange economy a game with only two players, regardless of the number of consumers. In this two-player game, each player represents a different role of the society, formed by all the individuals in the economy. Player 1 selects feasible allocations trying to make Pareto improvements. Player 2 chooses an alternative from the wider range of allocations that are feasible in the sense of Aubin. The set of Nash equilibria of our game is non-empty and our main result provides a characterization of Walrasian equilibria allocations as strong Nash equilibria of the associated society game.     

A note on allocations attainable through nash equilibria

This note reexamines the issue of attaining efficient resource allocations as noncooperative equilibria of a strategic game. Contrary to the spirit of earlier work on this subject, it is shown that there is an entire class of choice rules, aside from the Walrasian one, which are implementable in Nash equilibrium stategies.     

On Nash implementation of Walrasian equilibria in a market for pure indivisibles without money

This paper deals with the Walrasian property of Nash and strong equilibria of a specific strategic market game which refers to a pure exchange economy involving purely indivisible commodities and no money. The game is of sealed-bid auction type and it is shown that any Nash equilibrium at which no agent is in status quo is a strong equilibrium and implements a Walrasian equilibrium. Moreover, it appears that two modifications of the game's rules ensure that any strong equilibrium outcome is Walrasian. These results are identical to those obtained by Svensson for markets involving purely indivisible goods and money.     

Asymptotic Efficiency for Discriminatory Private Value Auctions

We consider discriminatory auctions for multiple identical units of a good. Players have private values, possibly for multiple units. None of the usual assumptions about symmetry of players' distributions over values or symmetry of equilibrium play are made. Because of this, equilibria will typically involve inefficient allocations. Equilibria also become very difficult to solve for. Using an approach which does not depend on explicit equilibrium calculations we show that such auctions become arbitrarily close to efficient as the number of players, and possibly the number of objects, grows large, and provide a simple characterization of limit equilibria.     

Perfect competition in an oligopoly (including bilateral monopoly)

We show that if limit orders are required to vary smoothly, then strategic (Nash) equilibria of the double auction mechanism yield competitive (Walras) allocations. It is not necessary to have competitors on any side of any market: smooth trading is a substitute for price wars. In particular, Nash equilibria are Walrasian even in a bilateral monopoly.     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

Walrasian equilibria as limits of noncooperative equilibria. Part II: Pure strategies

We examine the connection between Walrasian equilibria of a limit economy (with infinitesimal firms) and noncooperative (Cournot) equilibria of approximating finite economies (with significant firms). Nonconvex production sets, decreasing returns in the aggregate, and endogenous determination of the number of active firms are allowed. A Walrasian equilibrium is a limit of pure strategy noncooperative equilibria only if a condition (loosely analogous to downward sloping demand in the partial equilibrium constant returns to scale case) holds. The condition is also sufficient to guarantee the existence of a robust sequence of pure strategy noncooperative equilibria which converges to the Walrasian equilibrium.     

Mixed bundling auctions

We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling.     

Double implementation in economies with production technologies unknown to the designer

Summary. This paper considers double implementation of Walrasian allocations and Lindahl allocations in Nash and strong Nash equilibria for both private and public goods economies when preferences, initial endowments, production technologies, and coalition patterns are all unknown to the designer. It will be noted that the mechanisms presented here are feasible and continuous. In addition, unlike most mechanisms proposed in the literature, our mechanism works not only for three or more agents, but also for two-agent economies, and thus it is a unified mechanism which is irrespective of the number of agents. Received: March 12, 1998; revised version: March 12, 1998     

Private monitoring in auctions

We study infinitely repeated first-price auctions in which a bidder only learns whether or not he won the object. While repetition of the stage-game equilibrium is the unique Nash equilibrium in public strategies, with patient bidders there are simple Nash equilibria in private strategies that improve on bid rotation. Sequential rationality is appropriately captured by essentially perfect Bayesian equilibrium (EPBE), which ignores behavior after irrelevant histories. Our main result is the construction of EPBEa that improve upon bid rotation. Assuming symmetry, the exclusionary schemes of Skrzypacz and Hopenhayn [Tacit collusion in repeated auctions, J. Econ. Theory 114 (2004), 153–169], including asymptotically efficient ones, are supported as EPBEa.     

Contests with rank-order spillovers

This paper presents a unified framework for characterizing symmetric equilibrium in simultaneous move, two-player, rank-order contests with complete information, in which each player??s strategy generates direct or indirect affine ??spillover?? effects that depend on the rank-order of her decision variable. These effects arise in natural interpretations of a number of important economic environments, as well as in classic contests adapted to recent experimental and behavioral models where individuals exhibit inequality aversion or regret. We provide the closed-form solution for the symmetric Nash equilibria of this class of games, and show how it can be used to directly solve for equilibrium behavior in auctions, pricing games, tournaments, R&D races, models of litigation, and a host of other contests.     

Efficient Equilibria in the Voluntary Contributions Mechanism with Private Information

We characterize generally the Bayesian Nash equilibria of a voluntary contributions public goods game for two consumers with private information.The two consumers simultaneously make voluntary contributions to the public good, and the contributions are refunded if the total falls short of the cost of the public good. Several families of equilibria (step‐function, regular and semi‐regular) are studied. Necessary and sufficient conditions for regular and semi‐regular equilibrium allocations to be interim incentive efficient are derived. In the uniform distribution case we prove (i) the existence of an open set of incentive efficient regular equilibria when the cost of production is large enough and (ii) the existence of an open set of incentive efficient semi‐regular equilibria when the cost of production is low enough. Step‐function equilibra are proved to be interim incentive inefficient.     

On Nash-Implementation in the Presence of Withholding

In this paper we construct a completely feasible and continuous mechanism whose Nash allocations coincide with Lindahl allocations when both preferences and initial endowments are unknown to the designer and unreported endowments are withheld. This mechanism extends the mechanism of Hong by allowing for semi-positivity of endowments for private goods economies and the mechanism of Tian by allowing for any number of private goods for public goods economies. Thus our mechanism improves all the existing mechanisms that implement Walrasian or Lindahl allocations. Journal of Economic Literature Classification Numbers: C72, D6l, D78, H41.     

Learning to Learn, Pattern Recognition, and Nash Equilibrium

The paper studies a large class of bounded-rationality, probabilistic learning models on strategic-form games. The main assumption is that players “recognize” cyclic patterns in the observed history of play. The main result is convergence with probability one to a fixed pattern of pure strategy Nash equilibria, in a large class of “simple games” in which the pure equilibria are nicely spread along the lattice of the game. We also prove that a necessary condition for convergence of behavior to a mixed strategy Nash equilibrium is that the players consider arbitrarily long histories when forming their predictions.Journal of Economic LiteratureClassification Numbers: C72, D83.     

Symmetric Separating Equilibria in English Auctions

We characterize the set of perfect Bayesian equilibria in symmetric separating strategies in the model of English auctions given by P. R. Milgrom and R. J. Weber (1982, Econometrica50, 1089–1122). There is a continuum of such equilibria. The equilibrium derived by Milgrom and Weber is that in which bids are maximal. Only in the case of pure private values does a restriction to weakly undominated strategies select a unique equilibrium. This has important implications for empirical studies of English auctions, particularly outside the pure private values paradigm. Journal of Economic Literature Classification Numbers: D44, D82.     

Strong Equilibrium in Congestion Games

[6]introduced the class of congestion games and proved that they always possess a Nash equilibrium in pure strategies. Here we obtain conditions for the existence of a strong equilibrium in this class of games, as well as for the equivalence of Nash and strong equilibria. We also give conditions for uniqueness and for Pareto optimality of the Nash equilibrium. Except for a natural monotonicity assumption on the utilities, the conditions are expressed only in terms of the underlying congestion game form. It turns out that avoiding a certain type of bad configuration in the strategy spaces is essential to positive results.Journal of Economic LiteratureClassification Numbers: C71, C72, D62.     

The Second Welfare Theorem in Economies with Non‐Walrasian Markets

The standard version of the second welfare theorem assumes that market operations produce Walrasian outcomes. Therefore, if there are individuals who can manipulate prices, the conclusion of the second welfare theorem is questionable. In this paper, we address the decentralization of a Pareto‐optimal allocation, when markets are non‐Walrasian. Our objective in this paper is to develop a game which can implement Pareto‐optimal allocations as Nash equilibria of strategic exchange in markets. In this way, we develop a version of the second welfare theorem for economies where markets are strategic.     

Characterization and incentive compatibility of Walrasian expectations equilibrium in infinite dimensional commodity spaces

Summary. We consider a differential information economy with infinitely many commodities and analyze the veto power of the grand coalition with respect the ability of blocking non-Walrasian expectations equilibrium allocations. We provide two different Walrasian expectations equilibrium equivalence results. First by perturbing the initial endowments in a precise direction we show that an allocation is a Walrasian expectations equilibrium if and only if it is not privately dominated by the grand coalition. The second characterization deals with the fuzzy veto in the sense of Aubin but within a differential information setting. This second equivalence result provides a different characterization for the Walrasian expectations equilibrium and shows that the grand coalition privately blocks in the sense of Aubin any non Walrasian expectations equilibrium allocation with endowment participation rate arbitrarily close to the total initial endowment participation for every individual. Finally, we show that any no free disposal Walrasian expectations equilibria is coalitional Bayesian incentive compatible. Since the deterministic Arrow-Debreu-McKenzie model is a special case of the differential information economy model, one derives new characterizations of the Walrasian equilibria in economies with infinitely many commodities.Received: 29 October 2003, Revised: 24 February 2004, JEL Classification Numbers: D51, D82, D11. Correspondence to: Emma Moreno-GarcíaThe authors are grateful to an anonymous referee for his/her careful reading and helpful comments and suggestions.C. Hervés and E. Moreno acknowledge support by Research Grant BEC2000-1388-C04-01 (Ministerio de Ciencia y Tecnología and FEDER); and support by the Research Grant SA091/02 (Junta de Castilla y León).     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.