零和游戏与锥程序的等价性
On the Equivalence of Zero-Sum Games and Conic Programs
AI总结：
研究证明了零和游戏与锥程序的强对偶性等价性，建立了统一框架，并展示了如何通过对偶对求解博弈价值和纳什均衡。
摘要：
研究证明了极小极大定理与锥线性规划强对偶定理在大规模游戏和锥程序中的几乎等价性。先前关于线性规划与双人零和游戏的等价性结果被扩展到巴纳赫空间，并建立了统一框架，将双人零和游戏与锥线性规划统一起来。具体而言，对于每一个具有双线性收益函数且策略集代表凸锥基的零和游戏，极小极大等式成立，其游戏价值和纳什均衡可通过求解对偶对的锥程序来找到。反之，同一类游戏的极小极大定理“几乎总是”意味着锥线性规划的强对偶性。事实上，我们给出了依赖于游戏的严格可行性的特征，并证明极小极大等价于维勒定理的广义版本。几个已确立的游戏类别被嵌入到引入的模型中，包括（i）半无限的，（ii）半定的，（iii）量子的，（iv）时间依赖的，（v）多项式游戏，以及（vi）任何具有紧策略集的连续游戏的混合扩展。
英文摘要：
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.
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