Universality of Nash components

We show that Nash equilibrium components are universal for the collection of connected polyhedral sets. More precisely for every polyhedral set we construct a so-called binary game—a game where all players have two pure strategies and a common utility function with values either zero or one—whose success set (the set of strategy profiles where the maximal payoff of one is indeed achieved) is homeomorphic to the given polyhedral set. Since compact semi-algebraic sets can be triangulated, a similar result follows for the collection of connected compact semi-algebraic sets.We discuss implications of our results for the strategic stability of success sets, and use the results to construct a Nash component with index k for any fixed integer k.     

Adaptation and complexity in repeated games

The paper presents a learning model for two-player infinitely repeated games. In an inference step players construct minimally complex inferences of strategies based on observed play, and in an adaptation step players choose minimally complex best responses to an inference. When players randomly select an inference from a probability distribution with full support the set of steady states is a subset of the set of Nash equilibria in which only stage game Nash equilibria are played. When players make ‘cautious’ inferences the set of steady states is the subset of self-confirming equilibria with Nash outcome paths. When players use different inference rules, the set of steady states can lie between the previous two cases.     

Reputation with observed actions

Summary A single long-run player plays a fixed stage game (simultaneous orsequential move) against an infinite sequence of short-run opponents that play only once but can observe all past realized actions. Assuming that the probability distributions over types of long and short-run players have full support, we show that the long-run player can always establish a reputation for theStackelberg strategy and is therefore guaranteed almost his Stackelberg payoff in all Nash equilibria of the repeated game.The financial support of the National Science Foundation, Grant SES 90-7999, and of Consiglio Nazionale delle Ricerche is gratefully acknowledged. I wish to thank David Levine, Wolfgang Pesendorfer and Seminar Participants at UCLA, Universidad Carlos III de Madrid and University of Naples for useful discussions and suggestions.     

Catalog competition and Nash equilibrium in nonlinear pricing games

We model strategic competition in a market with asymmetric information as a noncooperative game in which each seller competes for a buyer of unknown type by offering the buyer a catalog of products and prices. We call this game a catalog game. Our main objective is to show that catalog games have Nash equilibria. The Nash existence problem for catalog games is particularly contentious due to payoff discontinuities caused by tie-breaking. We make three contributions. First, we establish under very mild conditions on primitives that no matter what the tie-breaking rule, catalog games are uniformly payoff secure, and therefore have mixed extensions which are payoff secure. Second, we show that if the tie-breaking rule awards the sale to firms which value it most (i.e., breaks ties in favor of firms which stand to make the highest profit), then firm profits are reciprocally upper semicontinuous (i.e., the mixed catalog game is reciprocally upper semincontinuous). This in turn implies that the mixed catalog game satisfies Reny’s condition of better-reply security—a condition sufficient for existence (Reny in Econometrica 67:1029–1056, 1999). Third, we show by example that if the tie-breaking rule does not award the sale to firms which value it most (for example, if ties are broken randomly with equal probability), then the catalog game has no Nash equilibrium. This paper was written while the second author was Visiting Professor, Centre d’Economie de la Sorbonne, Universite Paris 1, Pantheon-Sorbonne. The second author thanks CES and Paris 1, and in particular, Bernard Cornet and Cuong Le Van for their support and hospitality. The second author also thanks the C&BA and EFLS at the University of Alabama for financial support. Both authors are grateful to Monique Florenzano and to participants in the April 2006 Paris 1 NSF/NBER Decentralization Conference for many helpful comments on an earlier version of the paper. Finally, both authors are especially grateful to an anonymous referee whose thoughtful comments led to substantial improvements in the paper. Monteiro acknowleges the financial support of Capes-Cofecub 468/04.     

The structure of the Nash equilibrium sets of standard 2-player games

Summary. In this paper I study a class of two-player games, in which both players action sets are [0,1] and their payoff functions are continuous in joint actions and quasi-concave in own actions. I show that a no-improper-crossing condition is both necessary and sufficient for a finite subset A of to be the set of Nash equilibria of such a game.Received: 21 November 2002, Revised: 9 September 2004, JEL Classification Numbers: C65, C72.I am grateful to an editor of the journal and an anonymous referee for their very helpful comments. I also would like to thank the seminar participants at City University of Hong Kong, Georgia State University, Northwestern University, and Rice University.     

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.     

Stochastic uncoupled dynamics and Nash equilibrium

In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence of play (the period-by-period behavior as well as the long-run frequency) to Nash equilibria of the one-shot stage game, and present a number of possibility and impossibility results. Basically, we show that if in addition to random experimentation some recall, or memory, is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it suffices to recall the last two periods of play.     

Continuity of the first price auction Nash equilibrium correspondence

Summary. Despite the complexity of the first price auction in the general asymmetric case, analytical results have started to emerge in the literature. Authors have also searched to gain insights by computing numerical estimates of the equilibria for some probability distributions of the valuations. This paper proves that the Nash equilibrium of the first price auction depends continuously, for the weak topology, on the valuation distributions and thus brings robustness to the numerical results as well as some theoretical results. As an example of application, we disprove a conjecture of comparative statics. Received: February 1, 1999; revised version: July 27, 2001     

Uniqueness and comparative statics of Nash equilibrium in a tariff retaliation model

This paper considers uniqueness and comparative statics of Nash equilibrium of a tariff retaliation model. The approach to the problem is geometrical and reminiscent of the analysis for the free trade competitive equilibrium. If the countries have constant elasticity of substitution utility functions, some simple conditions can be used to prove uniqueness of the Nash equilibrium of the tariff retaliation game. The welfare effects of endowment changes are analyzed in terms of the standard terms of trade and volume of trade effects. If the elasticity of substitution of one of the countries is sufficiently high, immiserizing growth will not occur.      

Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49–73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N→∞, the expected number of Nash equilibria is approximately . Letting M=N→∞, the expected number of Nash equilibria is , where is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.     

The envelope theorem for locally differentiable Nash equilibria of finite horizon differential games

Envelope theorems are established for a ubiquitous class of finite horizon differential games. The theorems cover open-loop and feedback information patterns in which the corresponding Nash equilibria are locally differentiable with respect to the parameters of the game. Their relationship with extant envelope results is discussed and an application of them to a generalized capital accumulation game is provided. An important implication of the theorems is that, in general, the archetypal economic interpretation of the costate vector, namely, as the shadow value of the state vector along the Nash equilibrium, is valid for feedback Nash equilibria, but not for open-loop Nash equilibria.     

Cournot–Nash equilibria in continuum games with non-ordered preferences

In the usual framework of continuum games with externalities, we substantially generalize Cournot–Nash existence results [Balder, A unifying approach to existence of Nash equilibria, Int. J.Game Theory 24 (1995) 79–94; On the existence of Cournot–Nash equilibria in continuum games, J. Math. Econ. 32 (1999) 207–223; A unifying pair of Cournot–Nash equilibrium existence results, J. Econ. Theory 102 (2002) 437–470] to games with possibly non-ordered preferences, providing a continuum analogue of the seminal existence results by Mas-Colell [An equilibrium existence theorem without complete or transitive preferences, J. Math. Econ. 1 (1974) 237–246], Gale and Mas-Colell [An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15], Shafer and Sonnenschein [Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348], Borglin and Keiding [Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorems, J. Math. Econ. 3 (1976) 313–316] and Yannelis and Prabhakar [Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12 (1983) 233–245].     

On the convergence to the Nash bargaining solution for action-dependent bargaining protocols

We consider a non-cooperative multilateral bargaining game and study an action-dependent bargaining protocol, that is, the probability with which a player becomes the proposer in a round of bargaining depends on the identity of the player who previously rejected. An important example is the frequently studied rejector-becomes-proposer protocol. We focus on subgame perfect equilibria in stationary strategies which are shown to exist and to be efficient. Equilibrium proposals do not depend on the probability to propose conditional on the rejection by another player. We consider the limit, as the bargaining friction vanishes. In case no player has a positive probability to propose conditional on his rejection, each player receives his utopia payoff conditional on being recognized. Otherwise, equilibrium proposals of all players converge to a weighted Nash bargaining solution, where the weights are determined by the probability to propose conditional on one's own rejection.     

Stability of the cooperative equilibrium in N-person prisoners' dilemma with sequential moves

Summary. Nishihara [3] showed that N-person prisoners' dilemma has a cooperative Nash equilibrium, if the players decide their actions sequentially in the order determined by Nature under a certain information structure, and if each player's payoffs satisfy a certain inequality. This paper examines the stability of this cooperative equilibrium against two matters: players' slight mistakes and deviations by coalitions. The main results are as follows: (i) if the inequality on each player's payoffs strictly holds, then the cooperative equilibrium is a strictly proper equilibrium; (ii) if N≤3, and if full cooperation is Pareto efficient in N-person prisoners' dilemma, then the cooperative equilibrium is a strong Nash equilibrium; (iii) the cooperative equilibrium is in general a coalition-proof Nash equilibrium. Received: June 23, 1997; revised version: December 2, 1997     

Existence of Nash equilibria in finite extensive form games with imperfect recall: A counterexample

This paper provides an example showing that for finite extensive form games without perfect recall existence of Nash equilibria in behaviour strategies is not guaranteed in general.     

团队生产的纳什均衡及合作博弈空间

本文在霍姆斯特姆所做贡献的基础上认为团队生产问题在本质上等同于囚徒困境问题,并指出霍姆斯特姆的实施强制契约的解决方法存在逻辑漏洞,在现实社会中一般是无效的.本文依据社会经济人个人效用最大化的条件定义了团队生产的合作战略博弈空间,合作战略博弈空间的均衡极点(即个人效用最大化点)就是团队生产收益分配博弈的纳什均衡解.如果个人收益的总和等于集体收益的最优值,那么个人理性与集体理性之间就达到统一,处于帕累托最优状态,即不存在道德风险;反之,则说明存在道德风险.并给出了在团队生产中通过改变合作战略博弈空间达到帕累托最优的一个特例,指出了科学合理地改变合作战略博弈空间在社会经济发展中的重要作用     

Teaching Nash Equilibrium and Dominance: A Classroom Experiment on the Beauty Contest

Abstract: The authors' aim in this article was to show how the use of classroom experiments may be a good pedagogical tool to teach the Nash equilibrium (NE) concept. The basic game is a version of the beauty contest game (BCG), a simple guessing game in which repetition lets students react to other players' choices and converge iteratively to the equilibrium solution. The authors perform this experiment with undergraduate students with no previous training in game theory. After four rounds, they observe a clear decreasing tendency in the average submitted number in all groups. Thus, the findings show that by playing a repeated BCG, students quickly learn how to reach the NE solution.     

A learning-based model of repeated games with incomplete information

This paper tests a learning-based model of strategic teaching in repeated games with incomplete information. The repeated game has a long-run player whose type is unknown to a group of short-run players. The proposed model assumes a fraction of ‘short-run’ players follow a one-parameter learning model (self-tuning EWA). In addition, some ‘long-run’ players are myopic while others are sophisticated and rationally anticipate how short-run players adjust their actions over time and “teach” the short-run players to maximize their long-run payoffs. All players optimize noisily. The proposed model nests an agent-based quantal-response equilibrium (AQRE) and the standard equilibrium models as special cases. Using data from 28 experimental sessions of trust and entry repeated games, including 8 previously unpublished sessions, the model fits substantially better than chance and much better than standard equilibrium models. Estimates show that most of the long-run players are sophisticated, and short-run players become more sophisticated with experience.