Mixed Nash Equilibria for Continuous Games and Reverse Mathematics

Abstract

The program reverse mathematics seeks to identify the minimal axioms needed to prove theorems of ordinary mathematics. In this paper, we develop  the reverse mathematics of measure theory and apply these results to the reverse mathematics study of game theory. We address the technical issue of  how to develop product measures in reverse mathematics, including formalizing Fubini’s thoerem. We show that weak compactness of probability  measures on a compact space is equivalent to arithmetical comprehension over RCA0. The forward direction is again slightly technical. As an application  of these results, we prove in ACA0 that any continuous game has a mixed Nash equilibrium.