  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.
