Representation of finite action large games

A large game can be formalized as a probability distribution on the set of players’ characteristics or as a function from a measure space of players to the set of players’ characteristics. Given a game as a probability distribution on the set of players’ characteristics, a representation of the game is a function from a set of players to the set of players’ characteristics which induces the same distribution. It is shown that if the playoffs are continuous and there are only finite number of actions, then the set of Nash equilibria of any representation of a game induces essentially all the Cournot-Nash equilibrium distributions of the given game.