A Characterization of Subgame-Perfect Equilibrium Plays in Borel Games of Perfect Information

We provide a characterization of subgame-perfect equilibrium plays in a class of perfect information games where each player’s payoff function is Borel measurable and has finite range. The set of subgame-perfect equilibrium plays is obtained through a process of iterative elimination of plays. Extensions to games with bounded Borel measurable payoff functions are discussed. As an application of our results, we show that if every player’s payoff function is bounded and upper semicontinuous, then, for every positive epsilon, the game admits a subgame-perfect epsilon-equilibrium. As we do not assume that the number of players is finite, this result generalizes the corresponding result of Purves and Sudderth [24] [Purves RA, Sudderth WD (2011) Perfect information games with upper semicontinuous payoffs. Math. Oper. Res. 36(3):468–473].