The possibility of impossible stairways: Tail events and countable player sets

In classical game theory, players have finitely many actions and evaluate outcomes of mixed strategies using a von Neumann-Morgenstern utility function. Allowing a larger, but countable, player set introduces phenomena that are impossible in finite games: Even if players have identical payoffs (no conflicts of interest), (1) this payoff may be minimized in dominant-strategy equilibria, and (2) games so alike that even the consequences of unilateral deviations are the same, may have disjoint sets of payoff-dominant equilibria. Moreover, a class of games without (pure or mixed) Nash equilibria is constructed.