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

We consider a symmetric three-players zero-sum game with two strategic variables. Three players are Players A, B and C. Two strategic variables are ti and si, i = A;B;C. They are related by invertible functions. Using the minimax theorem by Sion (1958) and the fixed point theorem by Glicksberg (1952) we will show that Nash equilibria in the following four states are equivalent. 1. All players, Players A, B and C choose ti; i = A;B;C, (as their strategic variables). 2. Two players choose ti's, and one player chooses si. 3. One player chooses ti, and two players choose si's. 4. All players, Players A, B and C choose si; i = A;B;C.