Two-Person Additively-Separable Sum Games

Abstract: We consider a subclass of bimatrix games which we refer to as two person additively separable sum games, where the sum of the payoffs of the two players is additively separable. The payoff to the row player 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 payoff to the column player 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 interdependent components of the payoffs of the two players is assumed to be zero. We 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 two person additively separable sum game. Before proving the desired result concerning the equivalence of the two sets, we provided a simple proof of the existence of equilibrium for two person additively separable sum games, using the strong duality theorem and the complementary slackness theorem of linear programming.