Minimax theorem and Nash equilibrium of symmetric multi-players zero-sum game with two strategic variables

TL;DR

This paper proves the equivalence of Nash equilibria in symmetric multi-player zero-sum games with two related strategic variables, using the minimax theorem, regardless of the combination of variables chosen by players.

Contribution

It establishes the equivalence of Nash equilibria across different strategic variable configurations in symmetric multi-player zero-sum games.

Findings

Nash equilibria are equivalent when all players choose the same strategic variable.

Equivalence holds even when players choose different variables.

The minimax theorem underpins the proof of equilibrium equivalence.

Abstract

We consider a symmetric multi-players zero-sum game with two strategic variables. There are $n$ players, $n\geq 3$. Each player is denoted by $i$. Two strategic variables are $t_i$ and $s_i$, $i\in \{1, \dots, n\}$. They are related by invertible functions. Using the minimax theorem by \cite{sion} we will show that Nash equilibria in the following states are equivalent. 1. All players choose $t_i,\ i\in \{1, \dots, n\}$, (as their strategic variables). 2. Some players choose $t_i$'s and the other players choose $s_i$'s. 3. All players choose $s_i,\ i\in \{1, \dots, n\}$.