Extremal structures and symmetric equilibria with countable actions

Abstract

In this paper we show that a Cournot-Nash equilibrium distribution τ of an atomless anonymous game with countable actions is a symmetric equilibrium if and only if it is an extreme point of the set of all Cournot-Nash equilibrium distributions of the game with the same marginals as τ. This characterization allows us to show, as an application of the Krein-Milman theorem, that any particular Cournot-Nash equilibrium of such a game can be reallocated such that players with the same characteristics always take the same action, which is to say that it can be symmetrized. As a consequence of the usual result on the existence of distributional equilibria, we also obtain the existence of symmetric equilibria for the games under consideration. © 1995.