Two-Person Zero Sum Games with Random Rewards: A Linear Programming Approach

Abstract

We consider bi-matrix games between a row player and a column player. The row player’s pay-off depends on a part that is determined by the strategy choices of both players plus a part that depends solely on what is chosen by the column player. Similarly, the column player’s pay-off depends on a part that is determined by the strategy choices of both players plus a part that depends solely on what is chosen by the row player. The sum of the parts of the payoffs that depend on the strategies of both players is equal to zero. We show that an equilibrium exists and a strategy profile is an equilibrium for such a game if and only if it is an equilibrium for the two-person zero-sum game determined by the interdependent parts of the pay-off matrices. Our proof relies almost entirely on the well-known result considering existence of equilibrium for two-person zero-sum games and the complete characterization of the set of such equilibria by the solution of a pair of linear programming problems that are dual to each other.