Games of incomplete information, ergodic theory, and the measurability of equilibria

We present an example of a one-stage three-player game of incomplete information played on a sequence space {0, 1}(Z) such that the players' locally finite beliefs are conditional probabilities of the canonical Bernoulli distribution on {0, 1}(Z), each player has only two moves, the payoff matrix is determined by the 0-coordinate and all three players know that part of the payoff matrix pertaining to their own payoffs. For this example there are many equilibria (assuming the axiom of choice) but none that involve measurable selections of behavior by the players. By measurable we mean with respect to the completion of the canonical probability measure, e.g., all subsets of outer measure zero are measurable. This example demonstrates that the existence of equilibria is also a philosophical issue.