Strong Nash equilibria and mixed strategies
Abstract
In this paper we consider strong Nash equilibria, in mixed strategies, for finite games. Any strong Nash equilibrium outcome is Pareto efficient for each coalition. First, we analyze the twoplayer setting. Our main result, in its simplest form, states that if a game has a strong Nash equilibrium with full support (that is, both players randomize among all pure strategies), then the game is strictly competitive. In order to get our result we use the indifference principle fulfilled by any Nash equilibrium, and the classical KKT conditions (in the vector setting), that are necessary conditions for Pareto efficiency. Our characterization enables us to design a strongNashequilibriumfinding algorithm with complexity in Smoothed$\mathcal{P}$. So, this problemthat Conitzer and Sandholm [Conitzer, V., Sandholm, T., 2008. New complexity results about Nash equilibria. Games Econ. Behav. 63, 621641] proved to be computationally hard in the worst caseis generically easy. Hence, although the worst case complexity of finding a strong Nash equilibrium is harder than that of finding a Nash equilibrium, once small perturbations are applied, finding a strong Nash is easier than finding a Nash equilibrium. Next we switch to the setting with more than two players. We demonstrate that a strong Nash equilibrium can exist in which an outcome that is strictly Pareto dominated by a Nash equilibrium occurs with positive probability. Finally, we prove that games that have a strong Nash equilibrium where at least one player puts positive probability on at least two pure strategies are extremely rare: they are of zero measure.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.05629
 Bibcode:
 2015arXiv150205629B
 Keywords:

 Mathematics  Optimization and Control