Mixed Nash equilibria for continuous games and reverse mathematics

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 RCA(0). The forward direction is again slightly technical. As an application of these results, we prove in ACA(0) that any continuous game has a mixed Nash equilibrium.