Non-cooperative games with a continuum of players whose payoffs depend on summary statistics

Summary. We consider static non-cooperative games with a continuum of small players whose payoffs depend on their own actions and finitely many summary statistics of the aggregate strategy profile. We prove the existence of an equilibrium in pure strategies without any convexity restrictions on payoffs or the common action space. We show that this result applies to a broad class of monopolistic competition models. Received: April 13, 2001; revised version: December 18, 2001 RID="*" ID="*" The result in this paper generalizes a result in my PhD dissertation supervised by M. Ali Khan and Joe Harrington. I thank them for support and encouragement. I also thank Sung Kim, Bruce Nanney, Ashvin Rajan, Kali Rath, and an anonymous referee for comments. The usual disclaimer applies.     

Fault tolerance in large games

A Nash equilibrium is an optimal strategy for each player under the assumption that others play according to their respective Nash strategies, but it provides no guarantees in the presence of irrational players or coalitions of colluding players. In fact, no such guarantees exist in general. However, in this paper we show that large games are innately fault tolerant. We quantify the ways in which two subclasses of large games – λ-continuous games and anonymous games – are resilient against Byzantine faults (i.e. irrational behavior), coalitions, and asynchronous play. We also show that general large games have some non-trivial resilience against faults.     

Anonymity in nonatomic games

This paper purpose is twofold. First, it offers a critical review of the proofs of existence of pure strategy Nash Equilibria in nonatomic games. In particular, it focuses on the alternative ways of formalizing the critical assumption of anonymity. Second, the paper proves the existence of pure strategy Nash Equilibria by relaxing anonymity and allowing instead for “limited anonymity” (i.e. players’ decisions depend on the average strategy of a finite number of players’ subsets and not on the average strategy of the whole set of players). (JEL: C72, C79)     

Random belief equilibrium in normal form games

In defining random belief equilibrium (RBE) in finite, normal form games we assume a player's beliefs about others' strategy choices are randomly drawn from a belief distribution that is dispersed around a central strategy profile, the focus. At an RBE: (1) Each chooses a best response relative to her beliefs. (2) Each player's expected choice coincides with the focus of the other players' belief distributions. RBE provides a statistical framework for estimation which we apply to data from three experimental games. We also characterize the limit-RBE as players' beliefs converge to certainty. When atoms in the belief distributions vanish in the limit, not all limit-RBE (called robust equilibria) are trembling hand perfect Nash equilibria and not all perfect equilibria are robust.     

Congestion games with load-dependent failures: Identical resources

We define a new class of games, congestion games with load-dependent failures (CGLFs). In a CGLF each player can choose a subset of a set of available resources in order to try and perform his task. We assume that the resources are identical but that players' benefits from successful completion of their tasks may differ. Each resource is associated with a cost of use and a failure probability which are load-dependent. Although CGLFs in general do not have a pure strategy Nash equilibrium, we prove the existence of a pure strategy Nash equilibrium in every CGLF with nondecreasing cost functions. Moreover, we present a polynomial time algorithm for computing such an equilibrium.     

On games with identical equilibrium payoffs

Summary. This paper compares the sets of Nash, coalition- proof Nash and strong Nash equilibrium payoffs of normal form games which are closely related. We propose sufficient conditions for equivalent or closely related games to have identical sets of equilibrium payoffs. Received: April 23, 1999; revised version: November 23, 1999     

Cournot games with biconcave demand

Biconcavity is a simple condition on inverse demand that corresponds to the ordinary concept of concavity after simultaneous parameterized transformations of price and quantity. The notion is employed here in the framework of the homogeneous-good Cournot model with potentially heterogeneous firms. The analysis leads to unified conditions, respectively, for the existence of a pure-strategy equilibrium via nonincreasing best-response selections, for existence via quasiconcavity, and for the uniqueness of the equilibrium. The usefulness of the generalizations is illustrated in cases where inverse demand is either “nearly linear” or isoelastic. It is also shown that commonly made assumptions regarding large outputs are often redundant.     

Large newsvendor games

We consider a game, called newsvendor game, where several retailers, who face a random demand, can pool their resources and build a centralized inventory that stocks a single item on their behalf. Profits have to be allocated in a way that is advantageous to all the retailers. A game in characteristic form is obtained by assigning to each coalition its optimal expected profit. We consider newsvendor games with possibly an infinite number of newsvendors. We prove in great generality results about balancedness of the game, and we show that in a game with a continuum of players, under a nonatomic condition on the demand, the core is a singleton. For a particular class of demands we show how the core shrinks to a singleton when the number of players increases.     

Mixed-strategy equilibria and strong purification for games with private and public information

This paper shows the existence of mixed-strategy equilibria for games with private and public information under general conditions. Under the additional assumptions of finiteness of action spaces and diffuseness and conditional independence of private information, a strong purification result is obtained for the mixed strategies in such games. As a corollary, the existence of pure-strategy equilibria follows. I am very grateful to Yeneng Sun, Nicholas C. Yannelis and M. Ali Khan for helpful discussions and suggestions. I also wish to thank an anonymous referee whose comments led to many improvements in the paper.     

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.     

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.     

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.     

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].     

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.     

Union games: technological unemployment

Summary. Given a production economy, we define union games by considering strategic behavior of the suppliers of factors. We refer to the Nash equilibria of this game as union equilibria. We analyze situations where the unemployment of factors is supported as a union equilibrium. The degree of unemployment depends on technological conditions. This allows us to model a source of unemployment which differs from the usual sources provided in the literature. We state a limit result that demonstrates that, as the market power of unions decreases, the corresponding sequence of union equilibria converges to the Walrasian equilibrium, that is, to full employment of factors. We also provide some examples that illustrate the main results.Received: 21 October 2004, Revised: 14 December 2004, JEL Classification Numbers: D51, C72. Correspondence to: Emma Moreno-GarcíaE. Moreno acknowledges financial support from the Research Grant BEC2000-1388-C04-01 (Ministerio de Ciencia y Tecnología and FEDER). G. Fernández de Córdoba and E. Moreno acknowledge financial support from the Research Grant SA091/02 from Junta de Castilla y León. We are indebted to C. Alós-Ferrer, C. Pita, D. Anisi, J. A. Ortega, F. Jimeno, J. P. Torres-Martínez, M. Steinert and C. Hervés for helpful comments and insights. We are particularly grateful to T. Kehoe and an anonymous referee for suggestions that improved this paper.     

Iterated strict dominance in general games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS^*) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS^* is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS^* by means of a “stability” criterion, and offer a sufficient and necessary epistemic condition for IESDS^*. We show by an example that IESDS^* may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS^* preserves the set of Nash equilibria.     

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.     

Generic determinacy of Nash equilibrium in network-formation games

This paper proves the generic determinacy of Nash equilibrium in network-formation games: for a generic assignment of utilities to networks, the set of probability distributions on networks induced by Nash equilibria is finite.