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 show that computing a pure(-strategy) perfect or proper equilibrium is $\mathsf{PLS}$-complete in concise potential games in normal form. For pure perfect equilibria, we extend this result to general polytope games, which includes extensive-form games. 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 pure perfect equilibrium can be computed in polynomial time, strengthening the existing results for pure Nash equilibria. More broadly, we make a connection between strongly polynomial-time algorithms and efficient perturbed optimization using fractional interpolation. 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. As a byproduct, this significantly strengthens and simplifies a seminal result of Etessami and Yannakakis (FOCS '07). 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 $\mathsf{CLS}$. This provides a principled and practical way of refining the landscape of gradient descent in constrained optimization.

翻译：计算均衡精炼的复杂性一直是算法博弈论研究的核心问题，但在势博弈这一奠基性博弈类中该问题长期悬而未决；本文填补了这一根本性空白。我们首先证明：在简约型标准式势博弈中，计算纯（策略）完美均衡或恰当均衡是$\mathsf{PLS}$完全的。对于纯完美均衡，我们将此结果推广至包含扩展式博弈的一般多面体博弈。接着我们转向更具结构性的博弈类别，即对称网络拥堵博弈与对称拟阵拥堵博弈。针对这两类博弈，我们证明纯完美均衡可在多项式时间内计算完成，这强化了现有关于纯纳什均衡的结果。更广泛而言，我们建立了强多项式时间算法与基于分数插值的高效扰动优化之间的关联。另一方面，我们证实在某类势博弈中，完美均衡与纳什均衡的最佳响应路径长度存在指数级分离。最后针对混合策略，我们证明即使在3人标准式势博弈中，计算几何邻近完美均衡的点也需要双指数级微小扰动。作为推论，这显著强化并简化了Etessami与Yannakakis（FOCS '07）的开创性结果。另一方面，在多矩阵势博弈这一特殊情形中，我们证明均衡精炼适用于扰动梯度下降动力学，因而属于复杂度类$\mathsf{CLS}$。这为约束优化中梯度下降的景观精炼提供了原则性且实用的方法。