Complexity Results about Nash Equilibria

TL;DR

This paper establishes the computational hardness of various problems related to finding and counting Nash equilibria in different game settings, highlighting fundamental complexity barriers.

Contribution

It provides a unified reduction demonstrating NP-hardness and #P-hardness for key equilibrium existence and counting problems, even in simple or symmetric games.

Findings

NP-hardness of existence of Nash equilibria with certain properties

#P-hardness of counting Nash equilibria

NP-hardness of pure-strategy Bayes-Nash equilibrium existence

Abstract

Noncooperative game theory provides a normative framework for analyzing strategic interactions. However, for the toolbox to be operational, the solutions it defines will have to be computed. In this paper, we provide a single reduction that 1) demonstrates NP-hardness of determining whether Nash equilibria with certain natural properties exist, and 2) demonstrates the #P-hardness of counting Nash equilibria (or connected sets of Nash equilibria). We also show that 3) determining whether a pure-strategy Bayes-Nash equilibrium exists is NP-hard, and that 4) determining whether a pure-strategy Nash equilibrium exists in a stochastic (Markov) game is PSPACE-hard even if the game is invisible (this remains NP-hard if the game is finite). All of our hardness results hold even if there are only two players and the game is symmetric. Keywords: Nash equilibrium; game theory; computational complexity; noncooperative game theory; normal form game; stochastic game; Markov game; Bayes-Nash equilibrium; multiagent systems.