Nash equilibrium of partially asymmetric three-players zero-sum game with two strategic variables

TL;DR

This paper analyzes the Nash equilibria in a three-player zero-sum game with partial asymmetry and two strategic variables, revealing conditions under which different strategic choices lead to equivalent equilibria.

Contribution

It establishes the equivalence of equilibria under different strategic variable choices in a partially asymmetric three-player zero-sum game, extending symmetric game results.

Findings

Equilibrium with all players choosing $t_i$ is equivalent to a mixed strategy equilibrium.

Equilibrium with all players choosing $s_i$ is equivalent to a different mixed strategy equilibrium.

Equilibria with uniform strategies are not equivalent in the asymmetric case, unlike symmetric games.

Abstract

We consider a partially asymmetric three-players zero-sum game with two strategic variables. Two players (A and B) have the same payoff functions, and Player C does not. Two strategic variables are $t_i$'s and $s_i$'s for $i=A, B, C$. Mainly we will show the following results. 1. The equilibrium when all players choose $t_i$'s is equivalent to the equilibrium when Players A and B choose $t_i$'s and Player C chooses $s_C$ as their strategic variables. 2. The equilibrium when all players choose $s_i$'s is equivalent to the equilibrium when Players A and B choose $s_i$'s and Player C chooses $t_C$ as their strategic variables. The equilibrium when all players choose $t_i$'s and the equilibrium when all players choose $s_i$'s are not equivalent although they are equivalent in a symmetric game in which all players have the same payoff functions.