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We consider the relation between Sion's minimax theorem for a continuousfunction and a Nash equilibrium in a multi-players game with two groups whichis 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 impliesSion's minimax theorem with the coincidence of the maximin strategy and theminimax strategy for players in each group. %given the values of the strategicvariables.  2. Sion's minimax theorem with the coincidence of the maximin strategy andthe minimax strategy for players in each group implies the existence of a Nashequilibrium which is symmetric in each group.  Thus, they are equivalent. An example of such a game is a relative profitmaximization game in each group under oligopoly with two groups such that firmsin each group have the same cost functions and maximize their relative profitsin each group, and the demand functions are symmetric for the firms in eachgroup.

Sion's mini-max theorem and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group