Nash equilibrium in asymmetric multi-players zero-sum game with two strategic variables and only one alien

TL;DR

This paper analyzes a partially asymmetric multi-player zero-sum game with two strategic variables, showing conditions under which different equilibrium strategies are equivalent, highlighting differences from symmetric cases.

Contribution

It establishes the equivalence of equilibria under different strategic variable choices in asymmetric zero-sum games, extending prior 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 mixed strategy equilibrium.

Equilibria with all t_i and all s_i are not equivalent in asymmetric games.

Abstract

We consider a partially asymmetric multi-players zero-sum game with two strategic variables. All but one players have the same payoff functions, and one player (Player $n$) does not. Two strategic variables are $t_i$'s and $s_i$'s for each player $i$. Mainly we will show the following results. 1) The equilibrium when all players choose $t_i$'s is equivalent to the equilibrium when all but one players choose $t_i$'s and Player $n$ chooses $s_n$ as their strategic variables. 2) The equilibrium when all players choose $s_i$'s is equivalent to the equilibrium when all but one players choose $s_i$'s and Player $n$ chooses $t_n$ 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.