Symmetry, Equilibria, and Approximate Equilibria in Games With Countably Many Players

Abstract: I consider games with finite pure-strategy sets and countably many players. I present a “simple” example of such a game for which an ϵ-equilibrium exists for all ϵ > 0, but for which a Nash equilibrium does not exist. This game is not symmetric, which is inevitable in the following sense: under a mild condition on the utility function – the co-finiteness condition – existence of an ϵ-equilibrium for all ϵ > 0 in a symmetric game implies the existence of a Nash equilibrium in that game. The co-finiteness condition is logically unrelated to continuity.

Keywords: ϵ equilibrium, equilibrium non-existence, innite games, symmetry, tail events.

JEL Codes: C72.