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

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.