On the Equivalence of Zero-Sum Games and Conic Programs
证明了一大类零和博弈的极小极大定理与锥线性规划的强对偶定理几乎等价，将线性规划与零和博弈的经典等价性推广到巴拿赫空间，并建立了统一框架。
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.