Sion's mini-max theorem and Nash equilibrium in a five-players game with two groups which is zero-sum and symmetric in each group

TL;DR

This paper explores the equivalence between Sion's minimax theorem and symmetric Nash equilibria in a five-player, two-group zero-sum game, providing theoretical insights and an example related to oligopoly profit maximization.

Contribution

It establishes the equivalence between Sion's minimax theorem and symmetric Nash equilibria in a specific multi-player game setting, which was not previously known.

Findings

Symmetric Nash equilibrium implies Sion's minimax theorem.

Sion's minimax theorem implies symmetric Nash equilibrium.

The results are demonstrated with an oligopoly profit maximization example.

Abstract

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in a five-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 Sion's minimax theorem for a pair of playes in each group. 2. Sion's minimax theorem for a pair of playes in each group imply the existence of a 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.