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

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