Pure strategy Nash equilibria of large finite-player games and their relationship to non-atomic games

We consider Nash equilibria of large anonymous games (i.e., each player's payoff depends on his choice and the distribution of the choices made by others). We show that pure strategy Nash equilibria exist in all sufficiently large finite-player games with finite action spaces and for generic distributions of players' payoff functions. We also show that equilibrium distributions of non-atomic games are asymptotically implementable in terms of Nash equilibria of large finite-player games. Extensions of these results to games with general compact metric action spaces are provided.