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Sion's minimax theorem and Nash equilibrium of symmetric three-players zero-sum game

Satoh, Atsuhiro and Tanaka, Yasuhito (2018): Sion's minimax theorem and Nash equilibrium of symmetric three-players zero-sum game.

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Abstract

About a symmetric three-players zero-sum game we will show the following results.

1. A modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium.

2. The existence of a symmetric Nash equilibrium is proved by the modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy.

Thus, they are equivalent. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. If it is symmetric, the maximin strategies and the minimax strategies constitute a Nash equilibrium. However, without the coincidence of the maximin strategy and the minimax strategy there may exist an asymmetric equilibrium in a symmetric three-players zero-sum game.

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