A note on discontinuity and approximate equilibria in games with infinitely many players

Peleg (1969) showed that it is possible for a game with countably many players and finitely many pure strategies to have no Nash equilibrium. In his example not only Nash, but even perfect ϵ-equilibrium fails to exist. However, the example is based on tail utility functions, and these have infinitely many discontinuity points. I demonstrate non-existence of perfect ϵ-equilibrium under a milder form of discontinuity: I construct a game with countably many players, finitely many pure strategies and no perfect ϵ-equilibrium, in which one player has a utility function with a single discontinuity point, and the utility of every other player is not only continuous, but depends on finitely many coordinates.