EXTREMAL STRUCTURES AND SYMMETRICAL EQUILIBRIA WITH COUNTABLE ACTIONS

In this paper we show that a Cournot-Nash equilibrium distribution tau of an atomless anonymous game with countable actions is a symmetric e quilibrium if and only if it is an extreme point of the set of all Cou rnot-Nash equilibrium distributions of the game with the same marginal s as tau. This characterization allows us to show, as an application o f the Krein-Milman theorem, that any particular Cournot-Nash equilibri um of such a game can be reallocated such that players with the same c haracteristics always take the same action, which is to say that it ca n be symmetrized. As a consequence of the usual result on the existenc e of distributional equilibria, we also obtain the existence of symmet ric equilibria for the games under consideration.