Sion’s Minimax Theorem and Nash Equilibria of Symmetric Multi-Players Zero-Sum Games with Continuous Strategies
Keywords:
Multi-players zero-sum game, Nash equilibrium, Sion’s minimax theorem, Cournot oligopoly.Abstract
About a symmetric multi-players zero-sum game with continuous strategies we will show the following results.
(1) Amodified version of Sion’s minimax theorem with the coincidence of the maximin strategy and the minimaxstrategy 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 may not correspond to Nash equilibrium strategies. However, if it is symmetric, the maximin strategies and the minimax strategies constitute a Nash equilibrium.
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