Minimax Theory and its Applications 04 (2019), No. 2, 387--396
Copyright Heldermann Verlag 2019
Sion's Minimax Theorem and Nash Equilibria of Symmetric Multi-Players Zero-Sum Games with Continuous Strategies
Atsuhiro Satoh
Faculty of Economics, Hokkai-Gakuen University, Sapporo 062-8605, Japan
atsatoh@hgu.jp
Yasuhito Tanaka
Faculty of Economics, Doshisha University, Kyoto 602-8580, Japan
yatanaka@mail.doshisha.ac.jp
About a symmetric multi-players zero-sum game with continuous strategies we will show the following results:
(1) A modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium.
(2) The existence of a symmetric Nash equilibrium is proved by the modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy.
Thus, they are equivalent. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players may not correspond to Nash equilibrium strategies. However, if it is symmetric, the maximin strategies and the minimax strategies constitute a Nash equilibrium.
Keywords: Multi-players zero-sum game, Nash equilibrium, Sion's minimax theorem, Cournot oligopoly.
MSC: 91A06, 91B26
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