Satoh, Atsuhiro and Tanaka, Yasuhito
(2018):
*On zero-sum game formulation of non zero-sum game.*

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## Abstract

We consider a formulation of a non zero-sum n players game by an n+1 players zero-sum game. We suppose the existence of the n+1-th player in addition to n players in the main game, and virtual subsidies to the n players which is provided by the n+1-th player. Its strategic variable affects only the subsidies, and does not affect choice of strategies by the n players in the main game. His objective function is the opposite of the sum of the payoffs of the n players. We will show 1) The minimax theorem by Sion (Sion(1958)) implies the existence of Nash equilibrium in the n players non zero-sum game. 2) The maximin strategy of each player in {1, 2, ..., n} with the minimax strategy of the n+1-th player is equivalent to the Nash equilibrium strategy of the n players non zero-sum game. 3) The existence of Nash equilibrium in the n players non zero-sum game implies Sion's minimax theorem for pairs of each of the n players and the n+1-th player.

Item Type: | MPRA Paper |
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Original Title: | On zero-sum game formulation of non zero-sum game |

Language: | English |

Keywords: | zero-sum game, non zero-sum game, minimax theorem, virtual subsidy |

Subjects: | C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games |

Item ID: | 88976 |

Depositing User: | Yasuhito Tanaka |

Date Deposited: | 15 Sep 2018 07:15 |

Last Modified: | 26 Sep 2019 20:56 |

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. Sion, M. (1958), ``On general minimax theorems,'' Pacific Journal of Mathematics, 8, pp. 171-176. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/88976 |