Symmetric Approximate Equilibrium Distributions with Finite Support

We show that a distribution of a game with a continuum of players is an equilibrium distribution if and only if there exists a sequence of symmetric approximate equilibrium distributions of games with finite support that converges to it. Thus, although not all games have symmetric equilibrium distributions, this result shows that all equilibrium distributions can be characterized by symmetric distributions of simpler games (i.e., games with a finite number of characteristics).