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.

Perfect and proper equilibria in large games

This paper studies pure strategy perfect and proper equilibria for games with non-atomic measure spaces of players and infinitely many actions. A richness condition (nowhere equivalence) on the measure space of players is shown to be both necessary and sufficient for the existence of such equilibria. The limit admissibility of perfect and proper equilibria is also proved.

Arbitrage equilibria in large games with many commodities

Can identical goods sell at different prices in identical markets when people are perfectly mobile? We provide a formal account of strategic behaviour in large games with many commodities, and exhibit how it drives price dispersion at equilibrium. Interactions between agents are modelled using a Shapley–Shubik market game. We demonstrate the failure of the law of one price in this setup through a robust counterexample. The proposed model, and our findings, constitute an alternative and plausible explanation to some “anomalies” which routinely appear in a wide array of fields, ranging from banking, business economics, to international, and labour economics.

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.

Equilibria in Large Games with Strategic Complementarities

We study the existence and computation of equilibrium in large games with strategic complementarities. Importantly, our class of games allows to analyze eco- nomic problems without any aggregative structure. Using monotone operators (in stochastic dominance orders) dened on the space of distributions, we rst prove existence of the greatest and least distributional Nash equilibrium under dierent set of assumptions than those in the existing literature. In addition, we provide results on computable monotone distributional equilibrium comparative statics rel- ative to ordered perturbations of the parameters of our games that were previously only available for games with aggregative structure. We conclude by discussing the question of equilibrium uniqueness, as well as presenting applications of our results to models of social distance, large stopping games, keeping up with the Joneses but

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.

Equilibria in Large Games with Strategic Complementarities

Equilibria in Large Games with Strategic Complementarities

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.

On the existence of pure-strategy 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 [D. Schmeidler, Equilibrium points of nonatomic games, J. Stat. Phys. 4 (1973) 295–300], Rashid [S. Rashid, Equilibrium points of non-atomic games: Asymptotic results, Econ. Letters 12 (1983) 7–10], Mas-Colell [A. Mas-Colell, On a theorem by Schmeidler, J. Math. Econ. 13 (1984) 201–206], Khan and Sun [M. Khan, Y. Sun, Non-cooperative games on hyperfinite Loeb spaces, J. Math. Econ. 31 (1999) 455–492] and Podczeck [K. Podczeck, On purification of measure-valued maps, Econ. Theory 38 (2009) 399–418]. 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.

On the Existence of Pure-Strategy 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.

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 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.

Equilibria in large games with continuous procedures

We study existence of equilibria in large games where players use boundedly rational procedures. The equilibria are different from Nash equilibria; the difference persists even when players use procedures for which the observations gathered in any period are used to evaluate the payoff from different actions.