We study strong Nash equilibria in mixed strategies in finite games. A Nash equilibrium is strong if no coalition of players can jointly deviate so that all players in the coalition get strictly better payoffs. Our main result concerns games with two players and states that if a game admits a strong Nash equilibrium, then the payoff pairs in the support of the equilibrium lie on a straight line in the players’ utility space. As a consequence, the set of games that have a strong Nash equilibrium in which at least one player plays a mixed strategy has measure zero. We show that the same property holds for games with more than two players, already when no coalition of two players can profitably deviate. Furthermore, we show that, in contrast to games with two players, in a strong Nash equilibrium an outcome that is strictly Pareto dominated may occur with positive probability.

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强纳什均衡和混合策略

我们研究了有限博弈中混合策略中的强纳什均衡。如果没有参与者联盟可以共同偏离以使联盟中的所有参与者获得严格更好的收益，那么纳什均衡就是强的。我们的主要结果涉及两个玩家的博弈，并指出如果一个博弈承认强纳什均衡，那么支持均衡的收益对位于玩家效用空间中的一条直线上。因此，具有强纳什均衡（其中至少有一个参与者采用混合策略）的博弈集测度为零。我们表明，当没有两个玩家的联盟可以有利可图地偏离时，具有两个以上玩家的游戏具有相同的属性。此外，我们表明，与两个玩家的游戏相比，