On Finding Pure Nash Equilibria of Discrete Preference Games and Network Coordination Games

This paper deals with the complexity of the problem of computing a pure Nash equilibrium for discrete preference games and network coordination games beyond $O(\log n)$-treewidth and tree metric spaces. First, we estimate the number of iterations of the best response dynamics for a discrete preference game on a discrete metric space with at least three strategies. Second, we present a sufficient condition that we have a polynomial-time algorithm to find a pure Nash equilibrium for a discrete preference game on a grid graph. Finally, we discuss the complexity of finding a pure Nash equilibrium for a two-strategic network coordination game whose cost functions satisfy submodularity. In this case, if every cost function is symmetric, the games are polynomial-time reducible to a discrete preference game on a path metric space.