Games with continuous payoff functions and the problem of measurability

In noncooperative game theory, it is often taken for granted that expected payoffs are well-defined and independent of the integral representation. However, this need not be the case even if strategy spaces are compact and payoffs are bounded. In this paper, we establish general conditions under which the measurability requirements for working with expected payoffs are automatically met. We use our findings to enhance Glicksberg's equilibrium existence theorem and to rigorously construct the mixed extension of discontinuous games such as contests and auctions.