THE NONEXISTENCE OF SYMMETRICAL EQUILIBRIA IN ANONYMOUS GAMES WITH COMPACT ACTION SPACES

In an anonymous game the payoff of a player depends upon the player's own action and the action distribution of all the players. If the game is atomless and the set of actions is finite, or countably infinite a nd compact, then there is a symmetric equilibrium distribution. Furthe rmore, every equilibrium distribution can be symmetrized. This note pr ovides three examples to the effect that the conditions in these resul ts cannot be relaxed. The first example is a game with atoms and finit e actions which has no symmetric equilibrium distribution. In the seco nd example, the game is atomless, the action space is uncountable and not every equilibrium distribution can be symmetrized. The third examp le shows that a symmetric equilibrium distribution may not exist in an atomless game with the interval [-1, 1] as the action space. A genera l construction then exhibits that given any uncountable compact metric space, there is an atomless game over that space which has no symmetr ic equilibrium distribution. A sufficient condition for an equilibrium distribution to be symmetrized is also given.