Maximin and minimax strategies in two-players game with two strategic variables

We examine maximin and minimax strategies for players in two-players game with two strategic variables and . We consider two patterns of game; one is the x-game in which strategic variables of players are x's, and the other is the p-game in which strategic variables of players are p's. We call two players Players A and B, and will show that the maximin strategy and the minimax strategy in the x-game, and the maximin strategy and the minimax strategy in the p-game are all equivalent for each player. However, the maximin strategy for Player A and that for Player B are not necessarily equivalent, and they are not necessarily equivalent to their Nash equilibrium strategies in the x-game nor the p-game. But, in a special case, where the objective function of Player B is the opposite of the objective function of Player A, the maximin strategy for Player A and that for Player B are equivalent, and they constitute the Nash equilibrium both in the x-game and the p-game.