Satoh, Atsuhiro and Tanaka, Yasuhito (2018): The equivalence of minimax theorem and existence of Nash equilibrium in asymmetric threeplayers zerosum game with two groups.
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Abstract
We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in an asymmetric threeplayers zerosum game with two groups. Two players are in Group A, and they have the same payoff function and strategy space. One player, Player C, is in Group C. Then,
1. The existence of a Nash equilibrium, which is symmetric in Group A, implies Sion's minimax theorem for pairs of a player in Group A and Player C with symmetry in Group A.
2. Sion's minimax theorem for pairs of a player in Group A and Player C with symmetry in Group A implies the existence of a Nash equilibrium which is symmetric in Group A.
Thus, they are equivalent.
Item Type:  MPRA Paper 

Original Title:  The equivalence of minimax theorem and existence of Nash equilibrium in asymmetric threeplayers zerosum game with two groups 
Language:  English 
Keywords:  threeplayers zerosum game, two groups, Nash equilibrium, Sion's minimax theorem 
Subjects:  C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C72  Noncooperative Games 
Item ID:  87260 
Depositing User:  Yasuhito Tanaka 
Date Deposited:  16 Jun 2018 10:40 
Last Modified:  16 Jun 2018 10:41 
References:  tKindler, 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. Satoh, A. and Y. Tanaka (2018), ``Sion's minimax theorem and Nash equilibrium of symmetric threeplayers zerosum game,'' mimeograph. 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, pp. 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/87260 
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The equivalence of minimax theorem and existence of Nash equilibrium in asymmetric threeplayers zerosum game with two groups. (deposited 16 Jun 2018 10:34)
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