Complexity Results about Nash Equilibria

• DOI: 10.48550/arxiv.cs/0205074
• Publication Date: 2002-01-01
• Authors: Conitzer, Vincent; Sandholm, Tuomas
• Keywords: FOS: Computer and information sciences; Computer Science - Computational Complexity; I.2.11; Computer Science - Computer Science and Game Theory; 0202 electrical engineering, electronic engineering, information engineering; Computer Science - Multiagent Systems; 02 engineering and technology; Computational Complexity (cs.CC); Computer Science and Game Theory (cs.GT); Multiagent Systems (cs.MA)

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.