
Minimax Theory and its Applications 04 (2019), No. 2, 387396 Copyright Heldermann Verlag 2019 Sion's Minimax Theorem and Nash Equilibria of Symmetric MultiPlayers ZeroSum Games with Continuous Strategies Atsuhiro Satoh Faculty of Economics, HokkaiGakuen University, Sapporo 0628605, Japan atsatoh@hgu.jp Yasuhito Tanaka Faculty of Economics, Doshisha University, Kyoto 6028580, Japan yatanaka@mail.doshisha.ac.jp About a symmetric multiplayers zerosum 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 zerosum 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: Multiplayers zerosum game, Nash equilibrium, Sion's minimax theorem, Cournot oligopoly. MSC: 91A06, 91B26 [ Fulltextpdf (94 KB)] for subscribers only. 