On Equilibrium in Pure Stategies in Games with Many Players

We introduce a framework of noncooperative games, allowing both countable sets of pure strategies and player types, in which players are characterized by their attributes and demonstrate that for all games with su¢ciently many players, every mixed strategy Nash equilibrium can be used to construct a Nash "-equilibrium in pure strategies that is ‘"-equivalent’. Our framework introduces and exploits a distinction between crowding attributes of players (their external e¤ects on others) and their taste attributes (their payo¤ functions). The set of crowding attributes is assumed to be compact; this is not required, however, for taste attributes. For the special case of at most a ...nite number of crowding attributes, we obtain analogs, for ...nite games, of puri...cation results due to Pascoa (1993a,b,1998) for games with a continuum of players. Our main theorems are based on a new mathematical result, in the spirit of the Shapley-Folkman Theorem but applicable to a countable (not necessarily ...nite dimensional) strategy space. 1 Our research in the context of the literature on puri...cation Whether a non-cooperative equilibrium in mixed strategies can by approximated by a strategy vector in pure strategies – in other words, ‘puri...ed’ – as a consequence of large numbers of players was ...rst addressed by Schmeidler (1973) for a game with a continuum of players. Since then, many authors have contributed to this literature (including Mas-Colell 1984, Khan 1989, 1998, Khan et al. 1997, Pascoa 1998 and Khan and Sun 1999, for example). Games with a continuum of players are typically motivated as idealizations of games with a large but ...nite player set. In this paper we provide analogues,