Independence of existence of measurable equilibrium selections

Abstract We study the problem of selecting, in a measurable manner, equilibria from each of a family of games. Typical equilibrium selection theorems assume continuity in, and compactness of, actions, while we merely assume equilibria exist for all games in the family, and payoffs are jointly Borel in parameter and actions. The existence of Lebesgue- or universally measurable selectors turns out to be independent of ZFC; the result is robust to restriction to zero-sum games, as well as to allowing mixed strategies. We show, however, that the existence of analytically measurable selections, well-known to exist for single decision makers, fails for families of two-player zero-sum games.