Anti-coordination Games and Stable Graph Colorings

Motivated by understanding non-strict and strict pure strategy equilibria in network anti-coordination games, we define notions of stable and, respectively, strictly stable colorings in graphs. We characterize the cases when such colorings exist and when the decision problem is NP-hard. These correspond to finding pure strategy equilibria in the anti-coordination games, whose price of anarchy we also analyze. We further consider the directed case, a generalization that captures both coordination and anti-coordination. We prove the decision problem for non-strict equilibria in directed graphs is NP-hard. Our notions also have multiple connections to other combinatorial questions, and our results resolve some open problems in these areas, most notably the complexity of the strictly unfriendly partition problem.