Stability and equilibrium selection in a link formation game

In this paper we study a static link formation game under consent that has multiple Nash equilibria. In the literature, the use of coalitional refinements has been the standard approach to select among equilibria. Alternatively, based on the Global Games theory, a non cooperative equilibrium selection approach is proposed, so as to select those Nash equilibria that are robust to the introduction of incomplete information. Interestingly, the equilibrium selected is unique and it is in conflict with those predicted by the commonly used coalitional refinements. Furthermore, a conflict is found between stability and efficiency even when no such conflict exists with the coalitional refinements. We are especially grateful to Deborah Minehart, Roger Lagunoff, Daniel Vincent, Peter Cramton, Luca Anderlini, Axel Anderson, Felipe Zurita, Francis Bloch, Colin Stewart, the editor and three anonymous referees for their valuable comments that contributed to improve the paper. We also thank participants at the EEA-ESEM, Stockholm 2003, the IUSC conference, Columbia University 2003, and the Workshop on Global Games, SUNY, Stony Brook, 2007.     

Scale functions in equilibrium selection games

We present here an evolutionary game model, and address the issue of equilibrium selection working with the scale function of a diffusion process describing the dynamics of population processes with mutation modeled as white noise. This model is the same as the one in Foster and Young (1990) but with a different interpretation at the boundaries and with different mutation modelings. First, we justifiably assume that the boundaries of the solution of the stochastic differential equation are absorbing so that the first boundary of the interval [0,1] hit will determine the equilibrium selected. Then, working with the scale function, we obtain for 2×2 symmetric games and different mutation parameters, some new and interesting equilibrium selection results. The aim of this article is to describe another method of approach in evolutionary games with mutation which we believe will prove to be very useful in studying more general normal form games and different mutation modelings.     

Mistaken self-perception and equilibrium

Summary. Motivated by real-world information economics problems and by experimental findings on overconfidence, this paper introduces a general epistemic construction to model strategic interaction with incomplete information, where the players self-perception may be mistaken. This allows us to rigorously describe equilibrium play, by formulating appropriate equilibrium concepts. We show that there always exist objective equilibria, where the players correctly anticipate each others strategies without attempting to make sense of them, and that these outcomes coincide with the equilibria of an associated Bayesian game with subjective priors. In population games, these equilibria can also be always introspectively rationalized by the players, despite their possibly mistaken self-perception.Received: Received: May, 12, 2003; revised version: 3 December 2004, Revised: 3 December 2004, JEL Classification Numbers: J31, D82, D83.I thank Massimiliano Amarante, Eddie Dekel, Massimo Marinacci, Jean Francois Mertens, Giuseppe Moscarini, Paolo Siconolfi, Marciano Siniscalchi, Joel Sobel, and an anonymous referee.     

Ambiguity,uncertainty aversion and equilibrium welfare

Summary. In order to analyse the effect of ambiguity and uncertainty aversion on equilibrium welfare, a two period, pure exchange one good economy is considered. Agents are Choquet-expected-utility maximizers with same convex capacity and strictly concave utility index. It is proven that equilibrium is indeterminate whenever several probabilities in the core of the capacity minimize the expected value of aggregate endowment and not all agents have same expected endowments under those probabilities. It is further shown that small changes in aggregate endowment may have drastic welfare implications. A more general model is considered in the case of no aggregate uncertainty: agents have a set of priors and are uncertainty averse as modelled by Gilboa-Schmeidler [1989]. In the case of complete markets, it is shown that assets have a spread of equilibrium prices similar to the spread of no-arbitrage prices compatible with absence of arbitrage in markets with imperfections.Received: 2 June 2000, Revised: 27 March 2003, JEL Classification Numbers: D46, D59,D60, G12.I have benefited from conversations with L. Epstein, F. Magnien and J. M. Tallon.     

Equilibrium selection in coordination games with simultaneous play

Summary. We apply the dynamic stochastic framework proposed in recent evolutionary literature to a class of coordination games played simultaneously by the entire population. In these games payoffs, and hence best replies, are determined by a summary statistic of the population strategy profile. We demonstrate that with simultaneous play, the equilibrium selection depends crucially on how best responses to the summary statistic remain piece-wise constant. In fact, all the strict Nash equilibria in the underlying stage game can be made stochastically stable depending on how the best response mapping generates piece-wise constant best responses. Received: February 12, 2001; revised version: October 29, 2001     

Equilibrium selection in common-value second-price auctions

This note considers equilibrium selection in common-value second-price auctions with two bidders. We show that for each ex post equilibrium in continuous and undominated strategies, a sequence of “almost common-value” auctions can be constructed such that each of them possesses a unique undominated and continuous equilibrium and the corresponding sequence of equilibria converges to that ex post equilibrium. As an implication, no equilibrium selection of this model based on perturbations seems to be more convincing than others.     

The equilibrium manifold keeps the memory of individual demand functions

Summary. It is shown that the property that the equilibrium manifold keeps the memory of the individual demand functions holds true if every individual demand function satisfies the following three properties: 1) It is a function of commodity prices and of consumers income; 2) Consumption belongs to the nonnegative orthant of the commodity space; 3) Walras law. Neither differentiability nor continuity are necessary. In addition, the demand functions do not have to be utility maximizing subject to budget constraints.Received: 2 September 2003, Revised: 26 January 2004, JEL Classification Numbers: D50, D51.A preliminary version of this paper was released in October 1999 under the title Deriving individual demand functions from the equilibrium manifold. I wish to thank an anonymous referee for thoughtful comments.     

General equilibrium in economies with adverse selection

We model economies of adverse selection as Arrow–Debreu economies. In the spirit of Prescott and Townsend (Econometrica 52(1), 21–45, 1984a), we identify the consumption set of the individuals with the set of lotteries over net transfers. Thus, prices are linear in lotteries, but they may be non linear in commodity bundles. First, we study a weak equilibrium notion by viewing the economy of adverse selection as a pure exchange economy. The weak equilibrium set is non empty, but some of the allocations may be inefficient, and the equilibria indeterminate. Second, following Prescott and Townsend (Econometrica 52(1), 21–45, 1984a), we introduce an intermediary (firm) supplying feasible and incentive compatible measures. Equilibria are constrained efficient, but the equilibrium set is empty for an open set of economies containing the Rothschild and Stiglitz insurance economies. The research of A. Rustichini was supported by the NSF grant NSF/SES-0136556.     

Nash and Walras equilibrium via Brouwer

Summary. The existence of Nash and Walras equilibrium is proved via Brouwer's Fixed Point Theorem, without recourse to Kakutani's Fixed Point Theorem for correspondences. The domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasi-convex preferences. The key idea is to replace optimization with “satisficing improvement,” i.e., to replace the Maximum Principle with the “Satisficing Principle.” Received: July 9, 2001; revised version: February 25, 2002 RID="*" ID="*" I wish to thank Ken Arrow, Don Brown, and Andreu Mas-Colell for helpful comments. I first thought about using Brouwer's theorem without Kakutani's extension when I heard Herb Scarf's lectures on mathematical economics as an undergraduate in 1974, and then again when I read Tim Kehoe's 1980 Ph.D dissertation under Herb Scarf, but I did not resolve my confusion until I had to discuss Kehoe's presentation at the celebration for Herb Scarf's 65th birthday in September, 1995. RID="*" ID="*"Correspondence to: C. D. Aliprantis     

A note on the existence of a monetary equilibrium over an infinite horizon

Summary. This note considers an economy with ‘inside money’ that extends over an infinite horizon in the case of certainty. It shows the existence of an equilibrium and the indeterminacy of the overall price level when the supply of balances is set exogenously.Received: 6 November 2003, Revised: 2 September 2004 JEL Classification Numbers: D50, E40, E50.G. Bloise: I am indebted to Jacques Dréze, Leo Ferraris, Cuong Le Van, Luca Panaccione, Herakles Polemarchakis and Pietro Reichlin for the long discussions and their valuable suggestions. I also thank an anonymous referee for her/his advices.     

Continuity of the equilibrium price density and its uses in peak-load pricing

Summary. With as the commodity space, the equilibrium price density is shown to be a continuous function of the commodity characteristics. The result is based on symmetry ideas from the Hardy-Littlewood-Pólya theory of rearrangements. It includes, but is not limited to, the case of symmetric (rearrangement-invariant) production costs and additively separable consumer utility. Examples arise in continuous-time utility pricing, e.g., electricity pricing. In this context, a continuously varying price has two uses. First, it precludes demand jumps that would arise from discontinuous switches from one price rate to another. Second, in the problems of operating and valuing hydroelectric and pumped-storage plants (studied elsewhere), price continuity guarantees that their capacities (viz., the reservoir and the converter), the energy stocks, and in the case of hydro also the river flows, have well-defined marginal values.Received: 9 May 2001, Revised: 8 July 2004, JEL Classification Numbers: C62, D51, D58, L94. Correspondence to: Anthony HorsleyPart of this work (CentER DP 9014) was completed at the Center for Economic Research, Tilburg University, whose financial support is gratefully acknowledged. The extension to storage was supported by ESRC grant R000232822. We also thank the referee and the editor for their comments.     

Incentive monotonicity and equilibrium selection in 2×2 matrix games

For the class of 2×2 matrix games with two strict Nash equilibria the paper introduces an equilibrium refinement called incentive monotonicity. It selects the risk-dominant equilibrium if interests are conflicting, while it remains silent in games with common interests. These results suggest that the equilibrium-selection problem might be more difficult in games with common interests, which is certainly the case if risk dominance and payoff dominance go in opposite directions.     

Nash equilibrium in games with incomplete preferences

Summary. This paper investigates Nash equilibrium under the possibility that preferences may be incomplete. I characterize the Nash-equilibrium-set of such a game as the union of the Nash-equilibrium-sets of certain derived games with complete preferences. These games with complete preferences can be derived from the original game by a simple linear procedure, provided that preferences admit a concave vector-representation. These theorems extend some results on finite games by Shapley and Aumann. The applicability of the theoretical results is illustrated with examples from oligopolistic theory, where firms are modelled to aim at maximizing both profits and sales (and thus have multiple objectives). Mixed strategy and trembling hand perfect equilibria are also discussed.Received: 22 September 2003, Revised: 24 June 2004, JEL Classification Numbers: D11, C72, D43.I would like to thank Jean-Pierre Benôit, Juan Dubra, Alejandrio Jofre, Debraj Ray, Kim-Sau Chung and the seminar participants at NYU and at the Universidad de Chile for their comments. I am most grateful to Efe Ok, for his comments, criticism, suggestions and questions.     

Equilibrium selection through incomplete information in coordination games: an experimental study

We perform an experiment on a pure coordination game with uncertainty about the payoffs. Our game is closely related to models that have been used in many macroeconomic and financial applications to solve problems of equilibrium indeterminacy. In our experiment, each subject receives a noisy signal about the true payoffs. This game (inspired by the “global” games of Carlsson and van Damme, Econometrica, 61, 989–1018, 1993) has a unique strategy profile that survives the iterative deletion of strictly dominated strategies (thus a unique Nash equilibrium). The equilibrium outcome coincides, on average, with the risk-dominant equilibrium outcome of the underlying coordination game. In the baseline game, the behavior of the subjects converges to the theoretical prediction after enough experience has been gained. The data (and the comments) suggest that this behavior can be explained by learning. To test this hypothesis, we use a different game with incomplete information, related to a complete information game where learning and prior experiments suggest a different behavior. Indeed, in the second treatment, the behavior did not converge to equilibrium within 50 periods in some of the sessions. We also run both games under complete information. The results are sufficiently similar between complete and incomplete information to suggest that risk-dominance is also an important part of the explanation.      

Signaling in markets with two-sided adverse selection

Summary. The paper analyzes an economy with two-sided adverse selection, focusing on equilibria that satisfy a refinement based on the notion of strategic stability. In the familiar case of one-sided adverse selection, agents reveal all of their private information as long as the contract space is rich enough. However, with two-sided adverse selection, the sufficient conditions for separation are much stronger. Received: September 3, 1999; revised version: December 3, 1999     

Marshall on equilibrium and time: a reconstruction

This paper's aim is to offer a reconstruction of the Marshallian conception of equilibrium and time. Its main features are as follows. First, I argue that the hallmark of this conception is to posit an interrelationship between two equilibrium concepts — market-day and normal equilibrium. I claim that they are part and parcel and cannot be analysed separately. Second, my reconstruction gives a central role to the market period. Third, I argue that the so-called short and long-period equilibrium concepts refer to the same unique concept of normal equilibrium. Fourth, I argue that Marshall’s value theory admits the effective existence of disequilibrium states. A Marshallian disequilibrium refers to cases where market-day and normal values fail to coincide, this state of affairs going along, however, with market clearing. I also propose an alternative interpretation of Marshall's corn model wherein perfect information is considered the linchpin of achieving equilibrium. Finally, I argue that my reconstruction avoids a series of interpretative pitfalls.     

Equilibrium selection in bargaining models

This paper examines evolutionary equilibrium selection in bargaining models. We show that random best-response and continuous best-response learning dynamics give rise to (different) simple sufficient conditions for identifying outcomes as stochastically stable. This allows us to characterize the implications of these dynamics in simple bargaining games.     

Uniqueness of equilibrium in sealed high-bid auctions

For the case of two buyers we show that equilibrium in the sealed high-bid auction is unique when (i) buyers' reservations prices are drawn independently from distributions with finite support and positive mass at the lower endpoint; (ii) buyers have private values; and (iii) buyers' preferences are log supermodular. For more than two buyers, we obtain the same result under the additional assumptions that (iv) buyers with the same reservation price have the same preferences; (v) buyers are risk neutral or risk averse with non-increasing absolute risk aversion; and (vi) the supports of the different buyers' distributions of reservation prices have the same upper endpoint.     

Correlated quantal responses and equilibrium selection

This paper considers incomplete information games with payoffs subject to correlated random disturbances. It explains the connection between the uniqueness of quantal response equilibria, where large noise is required, and the uniqueness of equilibria in global games, where small noise is required.