On equilibrium in pure strategies in games with many players

Treating games of incomplete information with countable sets of actions and types and finite but large player sets we demonstrate that for every mixed strategy profile there is a pure strategy profile that is ‘εequivalent’. Our framework introduces and exploits a distinction between crowding attributes of players (their external effects on others) and their taste attributes (their payoff functions and any other attributes that are not directly relevant to other players). The main assumption is a ‘large game’ property,’ dictating that the actions of relatively small subsets of players cannot have large effects on the payoffs of others Since it is well known that, even allowing mixed strategies, with a countable set of actions a Nash equilibrium may not exist, we provide an existence of equilibrium theorem. The proof of existence relies on a relationship between the ‘better reply security’ property of Reny (1999) and a stronger version of the large game property. Our purification theorem are based on a new mathematical result, of independent interest, applicable to countable strategy spaces. 1 Motivation for the study of purification The concept of a Nash equilibrium is at the heart of much of economics and game theory. It is thus fundamental to question when Nash equilibrium provides a good description of human behavior. A number of challenges are posed by the evidence. Experimental evidence, for example, supports the view that individuals typically do not play mixed strategies (cf., Friedman 1996) and if they do, there may be serial correlation.1 Challenges are This has been demonstrated in a number of papers; see Walker and Wooders (2001) for a recent contribution and references therein.