Strong Nash equilibria and mixed strategies

• Published in: International journal of game theory Vol. 49; no. 3; pp. 699 - 710
• Main Authors: Braggion, Eleonora, Gatti, Nicola, Lucchetti, Roberto, Sandholm, Tuomas, von Stengel, Bernhard
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-09-2020
• Subjects: Behavioral/Experimental Economics; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Economics and Finance; Game Theory; Operations Research/Decision Theory; Original Paper; Social and Behav. Sciences; Strong Nash equilibrium; Noncooperative games; Mixed strategies; Pareto efficiency
• ISSN: 0020-7276; 1432-1270
• DOI: 10.1007/s00182-020-00723-3

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.