Pure Nash Equilibria: Hard and Easy Games

TL;DR

This paper explores the computational complexity of finding pure Nash equilibria in strategic games, showing NP-hardness in general but polynomial-time solutions under certain structural restrictions.

Contribution

It identifies structural conditions like small neighborhood and bounded hypertree width that enable efficient computation of pure Nash equilibria.

Findings

Determining pure Nash equilibria is NP-hard in general.

Certain structural restrictions allow polynomial-time solutions.

Graphical games with bounded treewidth are highly parallelizable.

Abstract

We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (the actions of) a logarithmically small number of other players. The dependency structure of a game G can be expressed by a graph DG(G) or by a hypergraph H(G). By relating Nash equilibrium problems to constraint satisfaction problems (CSPs), we show that if G has small neighborhood and if H(G) has bounded hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class NC2 of highly parallelizable problems.