On the relation between Sion's minimax theorem and existence of Nash equilibrium in asymmetric multi-players zero-sum game with only one alien

TL;DR

This paper explores the equivalence between Sion's minimax theorem and the existence of symmetric Nash equilibria in a specific asymmetric multi-player zero-sum game with one distinct player, establishing a bidirectional relationship.

Contribution

It demonstrates the equivalence between Sion's minimax theorem and symmetric Nash equilibrium existence in a particular class of asymmetric zero-sum games.

Findings

Nash equilibrium implies Sion's minimax theorem in the game setting.

Sion's minimax theorem implies the existence of symmetric Nash equilibrium.

The two concepts are shown to be equivalent in this context.

Abstract

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in an asymmetric multi-players zero-sum game in which only one player is different from other players, and the game is symmetric for the other players. Then, 1. The existence of a Nash equilibrium, which is symmetric for players other than one player, implies Sion's minimax theorem for pairs of this player and one of other players with symmetry for the other players. 2. Sion's minimax theorem for pairs of one player and one of other players with symmetry for the other players implies the existence of a Nash equilibrium which is symmetric for the other players. Thus, they are equivalent.