A saddle‐point theorem for a class of infinite games

In this paper, the existence of a saddle point for two‐person zero‐sum infinite games of a special type is proved. The games have continuous bilinear payoff functions and strategy sets which are convex, noncompact subsets of an infinite‐dimensional vector space. The closures of the strategy sets are, however, compact. The payoff functions satisfy conditions which allow the use of dominance arguments to show that points in the closure of a strategy set are dominated by or are strategically equivalent to points in the strategy set itself. Combining the dominance arguments with a well‐known existence theorem produces the main result of the paper. The class of games treated is an extension of a class studied by J. D. Matheson, who obtained explicit solutions for the saddle points by using necessary conditions.