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

We examine maximin and minimax strategies for players in symmetric multi-players game with two strategic variables. We consider two patterns of game; the x-game in which strategic variables of players are x's, and the p-game in which strategic variables of players are p's. We 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 for the players are all equivalent. However, the maximin strategy for the players 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 one player is the opposite of the sum of the objective functions of other players, the maximin and the minimax strategies for the players constitute the Nash equilibrium both in the x-game and the p-game.