Strong Nash equilibria and mixed strategies

TL;DR

This paper investigates strong Nash equilibria in mixed strategies for finite games, revealing conditions for their existence, computational complexity, and rarity, with implications for game theory and algorithm design.

Contribution

It characterizes strong Nash equilibria with full support, links them to strictly competitive games, and develops a more efficient algorithm for finding such equilibria.

Findings

Strong Nash equilibria with full support imply the game is strictly competitive.

Finding strong Nash equilibria is generically easier than worst-case complexity suggests.

Games with certain strong Nash equilibria are of zero measure, indicating rarity.

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 two--player 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 strong-Nash-equilibrium-finding algorithm with complexity in Smoothed-$\mathcal{P}$. So, this problem---that Conitzer and Sandholm [Conitzer, V., Sandholm, T., 2008. New complexity results about Nash equilibria. Games Econ. Behav. 63, 621--641] proved to be computationally hard in the worst case---is 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.