Solutions of Discrete, Two-Person Games

Abstract : This paper proposes to investigate the structure of solutions of discrete, zero-sum, two-person games. For a finite game-matrix it is well known that a solution (i.e., a pair of frequency distributions describing the optimal mixed strategies of the two players) always exists. Moreover, the set of solutions is known to be a convex polyhedron, each of whose vertices corresponds to a submatrix with special properties. In Part I of the paper a fundamental relationship between the dimensions of the sets of optimal strategies is proven, and devote particular attention to the set of games whose solutions are unique. Part II solves the problem of constructing a game-matrix with a given solution. A number of examples and geometrical arguments are interspersed to illustrate the theory, and Part III describes the solutions of some matrices with special diagonal properties.