  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 the
subsidies, and does not affect choice of strategies by the n players in the
main game. His objective function is the opposite of the sum of the payoffs of
the n players. We will show 1) The minimax theorem by Sion (Sion(1958)) implies
the existence of Nash equilibrium in the n players non zero-sum game. 2) The
maximin strategy of each player in {1, 2, \dots, n} with the minimax strategy
of the n+1-th player is equivalent to the Nash equilibrium strategy of the n
players non zero-sum game. 3) The existence of Nash equilibrium in the n
players non zero-sum game implies Sion's minimax theorem for pairs of each of
the n players and the n+1-th player.
