Subgame-perfect ϵ-equilibria in perfect information games with sigma-discrete discontinuities

Multi-player perfect information games are known to admit a subgame-perfect ϵ-equilibrium, for every ϵ>0, under the condition that every player’s payoff function is bounded and continuous on the whole set of plays. In this paper, we address the question on which subsets of plays the condition of payoff continuity can be dropped without losing existence. Our main result is that if payoff continuity only fails on a sigma-discrete set (a countable union of discrete sets) of plays, then a subgame-perfect ϵ-equilibrium, for every ϵ>0, still exists. For a partial converse, given any subset of plays that is not sigma-discrete, we construct a game in which the payoff functions are continuous outside this set but the game admits no subgame-perfect ϵ-equilibrium for small ϵ>0.