Pure Nash Equilibria: Complete Characterization of Hard and Easy Graphical Games

TL;DR

This paper characterizes exactly which classes of bounded in-degree graphs allow polynomial-time determination of pure Nash equilibria, linking tractability to the boundedness of treewidth in reduced graphs, and extends results to hypergraphical games.

Contribution

It provides a complete characterization of tractable classes of graphical games with bounded in-degree based on reduced graph treewidth, resolving a key open problem.

Findings

Pure Nash equilibrium decision problem is in P iff reduced graphs have bounded treewidth.

Characterization of tractable classes for colored hypergraphical games.

Novel use of Grohe's homomorphism complexity results.

Abstract

We consider the computational complexity of pure Nash equilibria in graphical games. It is known that the problem is NP-complete in general, but tractable (i.e., in P) for special classes of graphs such as those with bounded treewidth. It is then natural to ask: is it possible to characterize all tractable classes of graphs for this problem? In this work, we provide such a characterization for the case of bounded in-degree graphs, thereby resolving the gap between existing hardness and tractability results. In particular, we analyze the complexity of PUREGG(C, -), the problem of deciding the existence of pure Nash equilibria in graphical games whose underlying graphs are restricted to class C. We prove that, under reasonable complexity theoretic assumptions, for every recursively enumerable class C of directed graphs with bounded in-degree, PUREGG(C, -) is in polynomial time if and only if the reduced graphs (the graphs resulting from iterated removal of sinks) of C have bounded treewidth. We also give a characterization for PURECHG(C,-), the problem of deciding the existence of pure Nash equilibria in colored hypergraphical games, a game representation that can express the additional structure that some of the players have identical local utility functions. We show that the tractable classes of bounded-arity colored hypergraphical games are precisely those whose reduced graphs have bounded treewidth modulo homomorphic equivalence. Our proofs make novel use of Grohe's characterization of the complexity of homomorphism problems.