The Complexity of Equilibrium Refinements in Potential Games

The complexity of computing equilibrium refinements has been at the forefront of algorithmic game theory research, but it has remained open in the seminal class of potential games; we close this fundamental gap in this paper. We first establish that computing a pure-strategy perfect equilibrium is $\mathsf{PLS}$-complete under different game representations -- including extensive-form games and general polytope games, thereby being polynomial-time equivalent to pure Nash equilibria. For normal-form proper equilibria, our main result is that a perturbed (proper) best response can be computed efficiently in extensive-form games. As a byproduct, we establish $\mathsf{FIXP}_a$-completeness of normal-form proper equilibria in extensive-form games, resolving a long-standing open problem. In stark contrast, we show that computing a normal-form proper equilibrium in polytope potential games is both $\mathsf{NP}$-hard and $\mathsf{coNP}$-hard. We next turn to more structured classes of games, namely symmetric network congestion and symmetric matroid congestion games. For both classes, we show that a perfect pure-strategy equilibrium can be computed in polynomial time, strengthening the existing results for pure Nash equilibria. On the other hand, we establish that, for a certain class of potential games, there is an exponential separation in the length of the best-response path between perfect and Nash equilibria. Finally, for mixed strategies, we prove that computing a point geometrically near a perfect equilibrium requires a doubly exponentially small perturbation even in $3$-player potential games in normal form. On the flip side, in the special case of polymatrix potential games, we show that equilibrium refinements are amenable to perturbed gradient descent dynamics, thereby belonging to the complexity class $\mathsf{CLS}$.