  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.
