We prove the almost equivalence between two-player zero-sum games and conic linear programming problems in reflexive Banach spaces. The previous fundamental results of von Neumann, Dantzig, Adler, and von Stengel, regarding the equivalence between finite games with strategy sets defined over $\mathbb{R}^n$, and linear programming, are therefore generalized to the infinite-dimensional case. In fact, we show that for every two-player zero-sum game with a bilinear function of the form $u(x,y)=\langle y,Ax\rangle$, for some linear operator $A$, and strategy sets that represent bases of convex cones, the minimax theorem holds, and its game value and Nash equilibria can be computed by solving a primal-dual pair of conic linear problems. Conversely, the minimax theorem for the same class of games "almost always" implies strong duality of conic linear programming. The main results are applied to a number of infinite zero-sum games, whose classes include those of semi-infinite, semidefinite, time-continuous, quantum, polynomial, and homogeneous separable games.

翻译：我们证明了在自反巴拿赫空间中，双人零和博弈与锥线性规划问题几乎完全等价。冯·诺依曼、丹齐格、阿德勒和冯·施滕格尔关于定义在$\mathbb{R}^n$上有限博弈与线性规划等价性的经典结论，由此被推广至无限维情形。具体而言，我们证明：对于任意具有双线性形式$u(x,y)=\langle y,Ax\rangle$（其中$A$为线性算子）且策略集表示凸锥基的双人零和博弈，极小极大定理成立，其博弈值与纳什均衡可通过求解一对原始-对偶锥线性问题来计算。反之，对于同类博弈，极小极大定理"几乎总是"蕴含锥线性规划的强对偶性。主要结论被应用于若干类无限零和博弈，包括半无限、半定、时间连续、量子、多项式及齐次可分博弈。