An Ordinal Minimax Theorem

One of the earliest solution concepts considered in game theory are saddle points, combinations of actions such that no player can gain by deviating (see, e.g., von Neumann and Morgenstern, 1947). In two-player zero-sum games, every saddle point happens to coincide with the optimal outcome both players can guarantee in the worst case and thus enjoys a very strong normative foundation. Unfortunately, however, not every zero-sum game possesses a saddle point. This situation can be rectified by the introduction of mixed—i.e., randomized—strategies, as first proposed by Borel (1921). Von Neumann (1928) proved that every zero-sum game contains a mixed saddle point, or equilibrium, that moreover maintains two appealing properties of saddle points: interchangeability and equivalence. A zero-sum game may contain more than one equilibrium, but any combination of equilibrium strategies for either player forms an equilibrium (interchangeability), and all equilibria yield the same expected payoff (equivalence). Mixed equilibria have been criticized for resting on demanding epistemic assumptions such as the expected utility axioms by von Neumann and Morgenstern (1947). See, for example, Luce and Raiffa (1957, pp. 74–76) and Fishburn (1978). A compelling critique by Aumann addresses the use of mixed strategies in one-shot games: “When randomized strategies are used in a strategic game, payoff must be replaced by expected payoff. Since the game is played only once, the law of large numbers does not apply, so it is not clear why a player would be interested specifically in the mathematical expectation of his payoff” (Aumann, 1987, p. 63).