%0 Journal Article
%T Nash Equilibria in Large Games
%A Dionysius Glycopantis
%J Game Theory
%D 2014
%R 10.1155/2014/617596
%X 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. 1. Introduction This paper is in the context of ¡°large games.¡± First, we discuss the relationship of a particular Nash-Schmeidler, nonanonymous, formulation to a Mas-Colell anonymous game. We explain that, starting with the former, it is not always possible to define a Mas-Colell game. This is due to the different strategic behaviours of the players in the two formulations. The approach of Mas-Colell [1], although a general one, does not generalize that of Schmeidler [2]. The formulation of Schmeidler remains the more fundamental game theoretic approach. The description ¡°anonymous¡± refers to the fact that a player in Mas-Colell has no individual identity. What matters is the overall density of the choices, on which a set of possible utility functions depends. ¡°Nonanonymous¡± refers to the fact that in Schmeidler [2] a player is specified in terms of an individual utility function which depends on the specific actions of the opponents. The ideas above have an affinity to the issues discussed in the area of general equilibrium in markets with a continuum of traders. In the Schmeidler formulation, a game is defined as a measurable function from an atomless set of players to the set of players¡¯ characteristics. In the Mas-Colell alternative formulation, a game is defined anonymously, as a distribution on the players¡¯ characteristics. The relationship and the link between the two different formulations have been studied in the literature, under various assumptions, a number of times. Rath [3] uses the idea of the representation of a game through a measurable function which induces a given distribution. He then shows that for a finite number of actions the two formulations are, in terms of equilibria, essentially the same. Ali Khan and Sun [4], in their synthetic treatment, untangle in detail the relationship between the two formulations and show that only in the case of finite action spaces the two formulations are essentially the same. The two formulations have different implications when the set of actions
%U http://www.hindawi.com/journals/gt/2014/617596/