Pure Nash Equilibria in Graphical Games and Treewidth

Abstract

We treat PNE-GG, the problem of deciding the existence of a Pure Nash Equilibrium in a graphical game, and the role of treewidth in this problem. PNE-GG is known to be $$NP$$ -complete in general, but polynomially solvable for graphical games of bounded treewidth. We prove that PNE-GG is $$W[1]$$ -Hard when parameterized by treewidth. On the other hand, we give a dynamic programming approach that solves the problem in $$O^*(\alpha ^w)$$ time, where $$\alpha $$ is the cardinality of the largest strategy set and $$w$$ is the treewidth of the input graph (and $$O^*$$ hides polynomial factors). This proves that PNE-GG is in $$FPT$$ for the combined parameter $$(\alpha ,w)$$ . Moreover, we prove that there is no algorithm that solves PNE-GG in $$O^*((\alpha -\epsilon )^w)$$ time for any $$\epsilon > 0$$ , unless the Strong Exponential Time Hypothesis fails. Our lower bounds implicitly assume that $$\alpha \ge 3$$ ; we show that for $$\alpha =2$$ the problem can be solved in polynomial time. Finally, we discuss the implication for computing pure Nash equilibria in graphical games (PNE-GG) of $$O(\log n)$$ treewidth, the existence of polynomial kernels for PNE-GG parameterized by treewidth, and the construction of a sample and maximum-payoff pure Nash equilibrium.