General Theorem on a Finite Support of Mixed Strategy in the Theory of Zero-Sum Games

A theorem related to the theory of zero-sum games is proved. Rather general assumptions on the payoff function are found that are sufficient for an optimal strategy of one of the players to be chosen in the class of mixed strategies concentrated in at most m + 1 points if the opponent chooses a pure strategy in a finite-dimensional convex compact set and m is its dimension. This theorem generalizes results of several authors, starting from Bohnenblust, Karlin, and Shapley (1950).