The joint continuity of the expected payoff functions

Abstract

Glicksberg [Glicksberg, I.L., 1952. A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. In: Proceedings of the American Mathematical Society 3, pp. 170–174] generalized the Kakutani fixed point theorem to the setting of locally convex spaces and used it to prove that every k-person strategic game with action sets convex compact subsets of locally convex spaces and continuous payoff functions has a Nash equilibrium. He subsequently used this result to establish the following fundamental theorem of game theory: Every k-person strategic game with action sets metrizable compact topological spaces and continuous payoff functions has a mixed strategies equilibrium. However, in his proof of the latter result, Glicksberg did not show that the expected payoff functions were jointly continuous, something that was required for the existence of a mixed strategies equilibrium. The joint continuity of the expected payoff functions is not obvious at all. A general proof was presented by Glycopantis and Muir [D. Glycopantis, A. Muir, 2000. Continuity of the payoff functions, Economic Theory 16, 239–244]. An alternative proof was obtained by Zarichnyi [M. Zarichnyi, 2004. Continuity of the payoff function revisited, Economics Bulletin 3, 1–4]. A proof can also be obtained by using Billingsley [P. Billingsley, 1968. Convergence of Probability Measures, Wiley, New York, Theorem 3.2, p. 21] or by invoking a result of Balder [E.J. Balder, 1988. Generalized equilibrium results for games with incomplete information, Mathematics of Operations Research 13, 265–276, Theorem 3.1, p. 273]. In this note an alternative proof of the joint continuity of the expected payoff functions is given that is based on measure theoretic techniques.