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.

翻译：我们考虑一类双矩阵博弈的子类，称为二人可加可分和（TPASS）博弈，其中两名参与者的收益之和是可加可分的。行参与者在每一对纯策略下的收益是两个数字之和，其中第一个数字可能依赖于列参与者选择的纯策略，而第二个数字独立于列参与者选择的纯策略。列参与者在每一对纯策略下的收益也是两个数字之和，其中第一个数字可能依赖于行参与者选择的纯策略，而第二个数字独立于行参与者选择的纯策略。我们假设两名参与者收益中相互依赖的分量之和为零。我们证明了此类博弈均衡的存在性，并表明其均衡集是互为对偶的一对线性规划问题解在策略对集合上的投影。这一结果是二人零和博弈中相应著名结论的推广。我们还证明，一个（随机或混合）策略对是博弈的均衡当且仅当存在另外两个实数，使得三者共同满足某个线性规划问题。为证明此结果，我们需要引用TPASS博弈均衡的存在性。我们证明所用技术基于线性规划的对偶定理与互补松弛定理。