Nash Equilibria of Games with a Continuum of Players

We characterize Nash equilibria of games with a continuum of players (Mas-Colell (1984)) in terms of approximate equilibria of large finite games. For the concept of (ε, ε) – equilibrium — in which the fraction of players not ε – optimizing is less than ε — we show that a strategy is a Nash equilibrium in a game with a continuum of players if and only if there exists a sequence of finite games such that its restriction is an (εn, εn) – equilibria, with εn converging to zero. The same holds for ε – equilibrium — in which almost all players are ε – optimizing — provided that either players’ payoff functions are equicontinuous or players’ action space is finite. Furthermore, we give conditions under which the above results hold for all approximating sequences of games. In our characterizations, a sequence of finite games approaches the continuum game in the sense that the number of players converges to infinity and the distribution of characteristics and actions in the finite games converges to that of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance.N/