Satoh, Atsuhiro and Tanaka, Yasuhito (2017): Sion's minimax theorem and Nash equilibrium of symmetric multiperson zerosum game.
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Abstract
About a symmetric multiperson zerosum game we will show the following results.
1. The existence of a symmetric Nash equilibrium is proved by the Glicksberg fixed point theorem.
2. Sion's minimax theorem and the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium.
3. The existence of a symmetric Nash equilibrium is proved by Sion's minimax theorem and the coincidence of the maximin strategy and the minimax strategy.
If a zerosum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. However, if it is symmetric, the maximin strategies and the minimax strategies constitute a Nash equilibrium. With only the minimax theorem there may exist an asymmetric equilibrium in a symmetric multiperson zerosum game.
Item Type:  MPRA Paper 

Original Title:  Sion's minimax theorem and Nash equilibrium of symmetric multiperson zerosum game 
Language:  English 
Keywords:  multiperson zerosum game, Nash equilibrium, Sion's minimax theorem. 
Subjects:  C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C72  Noncooperative Games 
Item ID:  83484 
Depositing User:  Yasuhito Tanaka 
Date Deposited:  26 Dec 2017 08:49 
Last Modified:  09 Oct 2019 06:42 
References:  Glicksberg, I.L. (1952) ``A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points.'' Proceedings of the American Mathematical Society, 3, pp.170174. Kindler, J. (2005), ``A simple proof of Sion's minimax theorem,'' American Mathematical Monthly, 112, pp. 356358. Komiya, H. (1988), ``Elementary proof for Sion's minimax theorem,'' Kodai Mathematical Journal, 11, pp. 57. Matsumura, T., N. Matsushima and S. Cato (2013) ``Competitiveness and R&D competition revisited,'' Economic Modelling, 31, pp. 541547. Satoh, A. and Y. Tanaka (2013) ``Relative profit maximization and Bertrand equilibrium with quadratic cost functions,'' Economics and Business Letters, 2, pp. 134139, 2013. Satoh, A. and Y. Tanaka (2014a) ``Relative profit maximization and equivalence of Cournot and Bertrand equilibria in asymmetric duopoly,'' Economics Bulletin, 34, pp. 819827, 2014. Satoh, A. and Y. Tanaka (2014b), ``Relative profit maximization in asymmetric oligopoly,'' Economics Bulletin, 34, pp. 16531664. Sion, M. (1958), ``On general minimax theorems,'' Pacific Journal of Mathematics, 8, pp. 171176. Tanaka, Y. (2013a) ``Equivalence of Cournot and Bertrand equilibria in differentiated duopoly under relative profit maximization with linear demand,'' Economics Bulletin, 33, 14791486. Tanaka, Y. (2013b) ``Irrelevance of the choice of strategic variables in duopoly under relative profit maximization,'' Economics and Business Letters, 2, pp. 7583, 2013. VegaRedondo, F. (1997) ``The evolution of Walrasian behavior'' Econometrica, 65, pp. 375384. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/83484 
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Sion's minimax theorem and Nash equilibrium of symmetric multiperson zerosum game. (deposited 24 Oct 2017 14:07)
 Sion's minimax theorem and Nash equilibrium of symmetric multiperson zerosum game. (deposited 26 Dec 2017 08:49) [Currently Displayed]