Title: Sion's minimax theorem and Nash equilibrium of symmetric three-players zero-sum game

Authors: Atsuhiro Satoh; Yasuhito Tanaka

Addresses: Faculty of Economics, Hokkai-Gakuen University, Toyohira-ku, Sapporo, Hokkaido, 062-8605, Japan ' Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto, 602-8580, Japan

Abstract: About a symmetric three-players zero-sum game we will show the following results. 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. 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. However, without the coincidence of the maximin strategy and the minimax strategy there may exist an asymmetric equilibrium in a symmetric three-players zero-sum game.

Keywords: three-players zero-sum game; Nash equilibrium; Sion's minimax theorem.

DOI: 10.1504/IJMOR.2020.105859

International Journal of Mathematics in Operational Research, 2020 Vol.16 No.2, pp.279 - 289

Received: 23 Apr 2018
Accepted: 25 Dec 2018
Published online: 16 Mar 2020 *

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