On zero-sum game formulation of non zero-sum game

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

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
• 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

References

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