Treewidth and Pure Nash Equilibria

We consider the complexity of w-PNE-GG, the problem of computing pure Nash equilibria in graphical games parameterized by the treewidth w of the underlying graph. It is well-known that the problem of computing pure Nash equilibria is NP-hard in general, but in polynomial time when restricted to games of bounded treewidth. We now prove that w-PNE-GG is W[1]-hard. Next we present a dynamic programming approach, which in contrast to previous algorithms that rely on reductions to other problems, directly attacks w-PNE-GG. We show that our algorithm is in FPT for games with strategy sets of bounded cardinality. Finally, we discuss the implications for solving games of O(logn) treewidth, the existence of polynomial kernels for w-PNE-GG, and constructing a sample or a maximum-payoff pure Nash equilibrium.

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