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 on the equivalence between linear programming and finite games with strategy sets defined over $\mathbb{R}^n$, are therefore extended to more general strategy spaces. More specifically, we show that for every two-player zero-sum game with a bilinear payoff 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：我们在自反巴拿赫空间中证明了两人零和博弈与锥线性规划问题的近似等价性。因此，冯·诺依曼、丹齐格、阿德勒和冯·施滕格尔关于线性规划与策略集定义在$\mathbb{R}^n$上的有限博弈等价性的先前基本结果，被推广至更一般的策略空间。具体而言，我们证明对于每个两人零和博弈，若其具有形式为$u(x,y)=\langle y,Ax\rangle$的双线性支付函数（其中$A$为线性算子），且策略集表示凸锥的基，则极小极大定理成立，其博弈值及纳什均衡可通过求解一对锥线性规划的原-对偶问题来计算。反之，对于同一类博弈，极小极大定理"几乎总是"意味着锥线性规划的强对偶性。主要结果被应用于若干无限零和博弈，其类别包括半无限、半定、时间连续、量子、多项式及齐次可分离博弈。