Social Conformity and Equilibrium in Pure Strategies in Games with Many Players

We introduce a framework of noncooperative pregames, in which players are characterized by their attributes, and demonstrate that for all games with sufficiently many players, there exist approximate (ε) Nash equilibria in pure strategies. In fact, every mixed strategy equilibrium can be used to construct an ε-equilibrium in pure strategies, an ‘ε-purification’ result. Our main result is a social conformity theorem. Interpret a set of players, all with attributes in some convex subset of attribute space and all playing the same strategy, as a society. Observe that the number of societies may be as large as the number of players. Our social conformity result dictates that, given ε > 0, there is an integer L, depending on ε but not on the number of players, such that any sufficiently large game has an ε-equilibrium in pure strategies that induces a partition of the player set into fewer than L societies.