Satoh, Atsuhiro and Tanaka, Yasuhito (2018): The equivalence of mini-max theorem and existence of Nash equilibrium in asymmetric three-players zero-sum 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 three-players zero-sum 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 |
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Original Title: | The equivalence of mini-max theorem and existence of Nash equilibrium in asymmetric three-players zero-sum game with two groups |
Language: | English |
Keywords: | three-players zero-sum 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: | 87249 |
Depositing User: | Yasuhito Tanaka |
Date Deposited: | 16 Jun 2018 10:34 |
Last Modified: | 10 Oct 2019 20:33 |
References: | tKindler, J. (2005), ``A simple proof of Sion's minimax theorem,'' American Mathematical Monthly, 112, pp. 356-358. Komiya, H. (1988), ``Elementary proof for Sion's minimax theorem,'' Kodai Mathematical Journal, 11, pp. 5-7. Matsumura, T., N. Matsushima and S. Cato (2013) ``Competitiveness and R&D competition revisited,'' Economic Modelling, 31, pp. 541-547. Satoh, A. and Y. Tanaka (2013) ``Relative profit maximization and Bertrand equilibrium with quadratic cost functions,'' Economics and Business Letters, 2, pp. 134-139, 2013. Satoh, A. and Y. Tanaka (2014a) ``Relative profit maximization and equivalence of Cournot and Bertrand equilibria in asymmetric duopoly,'' Economics Bulletin, 34, pp. 819-827, 2014. Satoh, A. and Y. Tanaka (2014b), ``Relative profit maximization in asymmetric oligopoly,'' Economics Bulletin, 34, pp. 1653-1664. Satoh, A. and Y. Tanaka (2018), ``Sion's minimax theorem and Nash equilibrium of symmetric three-players zero-sum game,'' mimeograph. Sion, M. (1958), ``On general minimax theorems,'' Pacific Journal of Mathematics, 8, pp. 171-176. Tanaka, Y. (2013a) ``Equivalence of Cournot and Bertrand equilibria in differentiated duopoly under relative profit maximization with linear demand,'' Economics Bulletin, 33, pp. 1479-1486. Tanaka, Y. (2013b) ``Irrelevance of the choice of strategic variables in duopoly under relative profit maximization,'' Economics and Business Letters, 2, pp. 75-83, 2013. Vega-Redondo, F. (1997) ``The evolution of Walrasian behavior,'', Econometrica, 65, pp. 375-384. |
URI: | https://mpra.ub.uni-muenchen.de/id/eprint/87249 |
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