A Semi-Infinite Game

The ordinary finite, two-person, zero-sum game is completely defined by specifying an m × n game matrix A. The optimal strategies for both players, and the value of the game, can be obtained by solving a dual pair of linear programming problems. In this paper a semi-infinite game is defined; a semi-infinite game matrix has an infinite number of columns, i.e., the game is specified by a sequence of vectors {P_j} ∈ R^m. Optimal strategies and game values are shown to exist for the semi-infinite game by exploiting the relationship between these games and linear programming over cones.