On equivalent results in minimax theory
On equivalent results in minimax theory
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极小极大理论中的等价结果
DOI：
10.1016/j.ejor.2003.08.013
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发表时间：
2004
影响因子：
6.4
通讯作者：
J. Kolumbán
中科院分区：
文献类型：
--
作者：
J. B. G. Frenk;G. Kassay;J. Kolumbán
本文回顾了已知的极大极小定理及其在博弈论中的应用，并证明了这些定理可以用冯·诺伊曼1928年发表的有限策略集的二人零和博弈的第一极大极小定理来证明。这些结果包括Wald、Ville和Kneser的极大极小定理，以及Kakutani、Ky Fan、König、Neumann和Gwinner-Oettli对它们的推广。实际上，这些结果形成了一条等价链，这条等价链包含了两个不相交且其中一个紧致的闭凸集在有限维空间中的强分离结果。为了说明这一含义，作者仅使用紧集的简单性质和著名的weerstrass - lebesgue引理。
In this paper we review known minimax theorems with applications in game theory and show that these theorems can be proved using the first minimax theorem for a two-person zero-sum game with finite strategy sets published by von Neumann in 1928. Among these results are the well known minimax theorems of Wald, Ville and Kneser and their generalizations due to Kakutani, Ky Fan, König, Neumann and Gwinner–Oettli. Actually, it is shown that these results form an equivalent chain and this chain includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. To show the implications the authors only use simple properties of compact sets and the well-known Weierstrass–Lebesgue lemma.