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The concept of a mixed strategy, i.e., a probabilistic mixture of alternative courses of action, is of great theoretical importance in the mathematical analysis of conflict situations ("games"). The reason is related to the way in which "solutions" of a game are defined; the typical situation is that the theory guarantees that a game has a solution in terms of mixed strategies, but not necessarily in unmixed or "pure" strategies. In many prospective applications, however, the introduction of mixed strategies appears dubious at best , e.g., because adoption of such a strategy as modus operandi is simply not accept· able to the decision·maker concerned. This may for example reflect a natural identification of "probabilities" with "limiting relative frequencies of outcomes in indefinitely long sequences of repetitions of identical situations." For a decision-maker constrained to function in an everchanging environment, the notion of such sequences of perfect repetitions may be intolerable even as a hypothetical construct. Whatever the reason , this distaste for mixed strategies leads naturally to the question of assessing the likelihood that a game chosen "at random" will in fact possess a solution in pure strategies. Attention will be restricted here to noncooperative games with just two players; exten· sion of the results to the case of more than two players would be desirable, but at present is obstructed by combinatorial complications. The underlying model is formulated in section 2. Section 3 contains the derivations of two alternative formulas, (3.8) and (3.9), for the probability that a random two-player game (with a specified number of pure strategies for each player)fails to possess a solution in pure strategies. It is natural to inquire about the limiting behavior for large games, those in which both players have many alternative courses of action. In section 5 it is shown that the probability of a solution in pure strategies for large games is surprisingly far from negligible, in fact converging to

The probability of an equilibrium point