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

TL;DR

This paper presents a novel approach to non zero-sum n-player games by reformulating them as (n+1)-player zero-sum games, establishing links between Nash equilibria and minimax strategies through Sion's theorem.

Contribution

It introduces a new formulation connecting non zero-sum and zero-sum games, demonstrating the existence of Nash equilibria via minimax theorems and strategies.

Findings

Nash equilibrium existence follows from Sion's minimax theorem.

Maximin strategies align with Nash equilibria in the reformulation.

Nash equilibrium implies Sion's minimax theorem for each pair.

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, \dots, 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.