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We consider a formulation of a non zero-sum n players game by an n+1 playerszero-sum game. We suppose the existence of the n+1-th player in addition to nplayers in the main game, and virtual subsidies to the n players which isprovided by the n+1-th player. Its strategic variable affects only thesubsidies, and does not affect choice of strategies by the n players in themain game. His objective function is the opposite of the sum of the payoffs ofthe n players. We will show 1) The minimax theorem by Sion (Sion(1958)) impliesthe existence of Nash equilibrium in the n players non zero-sum game. 2) Themaximin strategy of each player in {1, 2, \dots, n} with the minimax strategyof the n+1-th player is equivalent to the Nash equilibrium strategy of the nplayers non zero-sum game. 3) The existence of Nash equilibrium in the nplayers non zero-sum game implies Sion's minimax theorem for pairs of each ofthe n players and the n+1-th player.

On zero-sum game formulation of non zero-sum game