Satoh, Atsuhiro and Tanaka, Yasuhito
(2018):
*Sion's mini-max theorem and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group.*

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## Abstract

We consider the relation between Sion's minimax theorem for a continuous function and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group. We will show the following results.

1. The existence of Nash equilibrium which is symmetric in each group implies a modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group. %given the values of the strategic variables.

2. A modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group implies the existence of Nash equilibrium which is symmetric in each group.

Thus, they are equivalent. An example of such a game is a relative profit maximization game in each group under oligopoly with two groups such that firms in each group have the same cost functions and maximize their relative profits in each group, and the demand functions are symmetric for the firms in each group.

Item Type: | MPRA Paper |
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Original Title: | Sion's mini-max theorem and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group |

Language: | English |

Keywords: | multi-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: | 88977 |

Depositing User: | Yasuhito Tanaka |

Date Deposited: | 15 Sep 2018 07:20 |

Last Modified: | 02 Oct 2019 01:05 |

References: | Kindler, 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. 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. Satoh, A. and Y. Tanaka (2014b), ``Relative profit maximization in asymmetric oligopoly,'' Economics Bulletin, 34, pp. 1653-1664. 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. Vega-Redondo, F. (1997) ``The evolution of Walrasian behavior,'' Econometrica, 65, pp. 375-384. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/88977 |