Two-Person Additively-Separable Sum Games

Abstract

We consider a sub-class of bi-matrix games which we refer to as two-person (hereafter referred to as two-player) additively-separable sum (TPASS) games, where the sum of the pay-offs of the two players is additively separable. The row player's pay-off at each pair of pure strategies, is the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the column player and the second being independent of the pure strategy chosen by the column player. The column player's pay-off at each pair of pure strategies, is also the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the row player and the second being independent of the pure strategy chosen by the row player. The sum of the inter-dependent components of the pay-offs of the two players is assumed to be zero. We prove the existence of equilibrium for such games and show that the set of equilibria for such games is the projection on the set of strategy pairs of the solutions of a pair of linear programming problems that are dual to each other. This result is a generalization of the corresponding and well-known result for two-person zero-sum games. We also show that a (randomized or mixed) strategy pair is an equilibrium of the game if and only if there exist two other real numbers such that the three together solve a certain linear programming problem. In order to prove this result, we need to appeal to the existence of an equilibrium for the TPASS game. The technology we use to prove our results, consists of the duality theorems and the complementary slackness theorem of linear programming.