We will show that Sion's minimax theorem is equivalent to the existence of Nash equilibrium in a symmetric multi-person zero-sum game. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. However, if it is...

About a symmetric multi-person zero-sum game we will show the following results. 1. The existence of a symmetric Nash equilibrium is proved by the Glicksberg fixed point theorem. 2. Sion's minimax theorem and the coincidence of the maximin strategy and the minimax strategy are proved by the...

About a symmetric multi-person zero-sum game we will show the following results. 1. Sion's minimax theorem plus the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium. 2. The existence of a symmetric Nash equilibrium is proved...

About a symmetric three-players zero-sum game we will show the following results. 1. A modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium. 2. The existence of a symmetric...

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...

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in an asymmetric three-players zero-sum game with two groups. Two players are in Group A, and they have the same payoff function and strategy space. One player, Player C, is in Group C. Then,...

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in an asymmetric three-players zero-sum game with two groups. Two players are in Group A, and they have the same payoff function and strategy space. One player, Player C, is in Group C. Then,...

We consider a formulation of a non zero-sum n players game by an n+1 players zero-sum game. We suppose the existence of the n+1-th player in addition to n players in the main game, and virtual subsidies to the n players which is provided by the n+1-th player. Its strategic variable affects only...

We consider the relation between Sion's minimax theorem for a continuous function and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group. We will show the following results. 1. The existence of Nash equilibrium which is symmetric in each group...

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...