Nash Equilibria as Limits of Equilibria of Nearby Finite Games

We study finite-player normal-form games with compact metric ac on spaces and bounded measurable payoffs. Our main theorem shows that every Nash equilibrium of such a game can be recovered as the limit, in the product weak topology, of Nash equilibria of finite games obtained by discre zing the ac on spaces and perturbing payoffs by a uniformly vanishing amount. The proof samples from the target equilibrium, uses concentra on inequali es to control weak convergence and incen ve constraints on a growing finite set, and then applies a payoff perturba on to convert the resul ng approximate equilibrium into an exact one. We also provide an example of a con nuous game with a Nash equilibrium that cannot be approximated through Nash equilibria of finite games without perturbing payoffs.