Nash Equilibria In Large Games Research Articles

On pure-strategy Nash equilibria in large games

A general condition called “coarser traits” is introduced and shown to be necessary and sufficient for the idealized limit property of large games with traits. Moreover, we illustrate the minimality of the “coarser traits” condition by showing its necessity in deriving the existence of pure-strategy Nash equilibria in large games with traits.

Approximation and characterization of Nash equilibria of large games

We characterize Nash equilibria of games with a continuum of players in terms of approximate equilibria of large finite games. This characterization precisely describes the relationship between the equilibrium sets of the two classes of games. In particular, it yields several approximation results for Nash equilibria of games with a continuum of players, which roughly state that all finite-player games that are sufficiently close to a given game with a continuum of players have approximate equilibria that are close to a given Nash equilibrium of the non-atomic game.

Characterization of Nash equilibria of large games

For a large game with traits, this paper characterizes its trait–action distributions in equilibrium by using an inequality under the following two sets of conditions: (i) both trait space and action space of the game contain at most countably many elements; (ii) the agent space is nowhere equivalent to the characteristic type space. Two examples are presented to show the failure of the characterization result when the conditions are not satisfied. Then, we also show the existence of the characterization inequality and the necessity of the two sets of conditions to the characterization results. Finally, the existence of Nash equilibria under the two sets of conditions comes naturally as a corollary.

Pure-strategy Nash equilibria in large games: characterization and existence

In this paper, we first characterize pure-strategy Nash equilibria in large games restricted with countable actions or countable payoffs. Then, we provide a counterexample to show that there is no such characterization when the agent space is an arbitrary atomless probability space (in particular, Lebesgue unit interval) and both actions and payoffs are uncountable. Nevertheless, if the agent space is a saturated probability space, the characterization result is still valid. Next, we show that the characterizing distributions for the equilibria exist in a quite general framework. This leads to the existence of pure-strategy Nash equilibria in three different settings of large games. Finally, we notice that our characterization result can also be used to characterize saturated probability spaces.

Nash Equilibria in Large Games

This paper adds to the discussion, in a general setting, that given a Nash-Schmeidler (nonanonymous) game it is not always possible to define a Mas-Colell (anonymous) game. In the two games, the players have different strategic behaviours and the formulations of the two problems are different. Also, we offer a novel explanation for the lack of a Nash equilibrium in an infinite game. We consider this game as the limit of a sequence of approximate, finite games for which an equilibrium exists. However, the limiting pure strategy function is not measurable.

Large games with a bio-social typology

We present a comprehensive theory of large games in which players have names and determinate social-types and/or biological traits, and identify through four decisive examples, essentially based on a matching-pennies type game, pathologies arising from the use of a Lebesgue interval for playerʼs names. In a sufficiently general context of traits and actions, we address this dissonance by showing a saturated probability space as being a necessary and sufficient name-space for the existence and upper hemi-continuity of pure-strategy Nash equilibria in large games with traits. We illustrate the idealized results by corresponding asymptotic results for an increasing sequence of finite games.

On the existence of Nash equilibria in large games

We study the existence of Nash equilibria in games with an infinite number of players. We show that there exists a Nash equilibrium in mixed strategies in all normal form games such that pure strategy sets are compact metric spaces and utility functions are continuous. The player set can be any nonempty set.

Existence of Nash equilibria in large games

Podczeck [Podczeck, K., 1997. Markets with infinitely many commodities and a continuum of agents with non-convex preferences. Economic Theory 9, 385–426] provided a mathematical formulation of the notion of “many economic agents of almost every type” and utilized this formulation as a sufficient condition for the existence of Walras equilibria in an exchange economy with a continuum of agents and an infinite dimensional commodity space. The primary objective of this article is to demonstrate that a variant of Podczeck’s condition provides a sufficient condition for the existence of pure-strategy Nash equilibria in a large non-anonymous game G when defined on an atomless probability space ( T , T , λ ) not necessary rich, and equipped with a common uncountable compact metric space of actions A. We also investigate to see whether the condition can be applied as well to the broader context of Bayesian equilibria and prove an analogue of Yannelis’s results [Yannelis, N.C., in press. Debreu’s social equilibrium theorem with asymmetric information and a continuum of agents. Economic Theory] on Debreu’s social equilibrium theorem with asymmetric information and a continuum of agents.

Purification of Bayesian–Nash equilibria in large games with compact type and action spaces

We present a purification result for incomplete information games with a large but finite number of players that allows compact metric spaces for both actions and types. We then compare our framework and findings to the early purification theorems of Rashid (1983. Equilibrium points of non-atomic games: asymptotic results. Economics Letters 12, 7–10), Cartwright and Wooders (2002 On equilibrium in pure strategies in games with many players. University of Warwick Working Paper 686 (revised 2005)), Kalai (2004. Large robust games. Econometrica 72, 1631–1665) and Wooders, Cartwright and Selten (2006. Behavioral conformity in games with many players. Games and Economic Behavior 57, 347–360). Our proofs are elementary and rely on the Shapley–Folkman theorem.

A Unifed Approach to the Purification of Nash Equilibria in Large Games

We present a purification result for incomplete information games with a large finite number of players that allows for compact metric spaces of actions and types. This result is then used to generalize the purfication theorems of Schmeidler (1973), Rashid (1983) and Kalai (2004). Our proofs are elementary and rely on the Shapley-Folkman theorem.

On the purification of Nash equilibria of large games

We consider Salim Rashid's asymptotic version of David Schmeidler's theorem on the purification of Nash equilibria. We show that, in contrast to what is stated, players' payoff functions have to be selected from an equicontinuous family in order for Rashid's theorem to hold. That is, a bound on the diversity of payoffs is needed in order for such asymptotic results to be valid.

On the Existence of Pure Strategy Nash Equilibria in Large Games

Over the years, several formalizations and existence results for games with a continuum of players have been given. These include those of Schmeidler (1973), Rashid (1983), Mas-Colell (1984), Khan and Sun (1999) and Podczeck (2007a). The level of generality of each of these existence results is typically regarded as a criterion to evaluate how appropriate is the corresponding formalization of large games. In contrast, we argue that such evaluation is pointless. In fact, we show that, in a precise sense, all the above existence results are equivalent. Thus, all of them are equally strong and therefore cannot rank the different formalizations of large games.

Some Refinements of Nash Equilibria of Large Games

The notions of perfect, proper, and persistent equilibria for large games are introduced. It is shown that every large game has at least one perfect and proper equilibrium. An example is given to show that a persistent equilibrium may fail to exist in a game. Journal of Economic Literature Classification Number: C79.