We prove the almost equivalence of the minimax theorem and the strong duality theorem for a large class of games and conic programs. The previous fundamental results on the equivalence of linear programming and two-player zero-sum games with simplex-strategy sets are extended to Banach spaces, and a comprehensive framework unifying two-player zero-sum games and conic linear programs is established. Specifically, we show that for every zero-sum game with a bilinear payoff function and strategy sets that represent bases of convex cones, the minimax equality holds and its game value and Nash equilibria can be found by solving a primal-dual pair of conic programs. Conversely, the minimax theorem for the same class of games "almost always" implies strong duality of conic linear programming. In fact, we give a game-dependent characterization of strict feasibility, and show that minimax is equivalent to a generalized version of Ville's theorem of the alternative. Several well-established game classes are embedded in the introduced model, including (i) semi-infinite, (ii) semidefinite, (iii) quantum, (iv) time-dependent, and (v) polynomial games, as well as (vi) the mixed extension of any continuous game with compact strategy sets.

翻译：我们证明了对于一大类博弈和锥规划问题，极小极大定理与强对偶定理几乎等价。先前关于线性规划与单纯形策略集下双人零和博弈等价性的基础性结论被扩展至巴拿赫空间，并建立了一个统一双人零和博弈与锥线性规划的综合框架。具体而言，我们证明：对于任意具有双线性收益函数且策略集为凸锥基集的零和博弈，其极小极大等式成立，且博弈值及纳什均衡可通过求解一对原始-对偶锥规划得到。反之，针对同类博弈的极小极大定理"几乎总是"蕴含锥线性规划的强对偶性。实际上，我们给出了严格可行性的博弈依赖性刻画，并证明极小极大等价于维莱择一性定理的广义形式。若干经典博弈类均被嵌入所提出的模型，包括：(i)半无限博弈、(ii)半定博弈、(iii)量子博弈、(iv)时变博弈、(v)多项式博弈，以及(vi)任意紧致策略集连续博弈的混合扩展。