The equivalence between von Neumann's Minimax Theorem for zero-sum games and the LP Duality Theorem connects cornerstone problems of the two fields of game theory and optimization, respectively, and has been the subject of intense scrutiny for seven decades. Yet, as observed in this paper, the proof of the difficult direction of this equivalence is unsatisfactory: It does not assign distinct roles to the two players of the game, as is natural from the definition of a zero-sum game. In retrospect, a partial resolution to this predicament was provided in another brilliant paper of von Neumann, which reduced the assignment problem to zero-sum games. However, the underlying LP is highly specialized; all entries of its objective function vector are strictly positive, the constraint vector is all ones, and the constraint matrix is 0/1. We generalize von Neumann's result along two directions, each allowing negative entries in certain parts of the LP. Our reductions make explicit the roles of the two players of the reduced game, namely their maximin strategies are to play optimal solutions to the primal and dual LPs. Furthermore, unlike previous reductions, the value of the reduced game reveals the value of the given LP. Our generalizations encompass several basic economic scenarios.

翻译：零和博弈的冯·诺依曼极小极大定理与线性规划对偶定理之间的等价性，分别连接了博弈论和优化领域中的核心问题，并已成为七十年来深入研究的主题。然而，正如本文所指出的，该等价性中困难方向的证明并不令人满意：它未能按照零和博弈定义的自然方式，为博弈的两位参与者分配不同的角色。回顾过去，冯·诺依曼在另一篇杰出论文中通过将指派问题约化为零和博弈，为这一困境提供了部分解决方案。然而，其所基于的线性规划具有高度特殊性：目标函数向量的所有分量均为严格正数，约束向量全为1，且约束矩阵为0/1矩阵。我们沿两个方向推广了冯·诺依曼的结果，每个方向都允许线性规划的特定部分出现负分量。我们的约化方法明确了约化后博弈中两位参与者的角色——即他们的极大极小策略对应于原始线性规划及其对偶规划的最优解。此外，与以往的约化方法不同，约化后博弈的值直接揭示了给定线性规划的最优值。我们的推广涵盖了几种基本的经济场景。