Nash Equilibria as Limits of Equilibria of Nearby Finite Games

We study finite-player normal-form games with compact metric action 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 discretizing the action spaces and perturbing payoffs by a uniformly vanishing amount. The proof samples from the target equilibrium, uses concentration inequalities to control weak convergence and incentive constraints on a growing finite set, and then applies a payoff perturbation to convert the resulting approximate equilibrium into an exact one. We also provide an example of a continuous game with a Nash equilibrium that cannot be approximated through Nash equilibria of finite games without perturbing payoffs.