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

We consider a symmetric multi-players zero-sum game with two strategic variables. There are $n$ players, $n\ge 3$. 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 $s_i,i\in \{1,\dots ,n\}$.