The Complexity of Equilibrium Refinements in Potential Games

2025-11-06

AI Summary

This study systematically characterizes the computational complexity of equilibrium refinements—specifically, perfect and proper equilibria—in potential games. Using techniques from algorithmic game theory, PLS/FIXPₐ complexity analysis, perturbed response dynamics, and best-response path modeling, it fully resolves this longstanding problem. It establishes that computing a pure-strategy perfect equilibrium is PLS-complete, while computing a normal-form proper equilibrium is FIXPₐ-complete. Polynomial-time algorithms are identified for symmetric network and matroid congestion games. The work further reveals that achieving a proper equilibrium in mixed strategies requires doubly exponentially small perturbations, and refutes the conjectured polynomial bound on the length of paths between perfect and Nash equilibria—demonstrating that such paths can be exponentially long. Finally, it proposes a perturbed gradient descent method that efficiently approximates equilibrium solutions in multi-matrix potential games.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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 $mathsf{CLS}$.

Problem

• Computing pure-strategy perfect equilibrium is PLS-complete in potential games
• Normal-form proper equilibrium computation is FIXP_a-complete in extensive-form games
• Perfect equilibrium computation shows exponential separation from Nash equilibria

Innovation

• Computing perfect equilibrium is PLS-complete in potential games
• Efficient proper best response computation in extensive-form games
• Polynomial-time perfect equilibrium in symmetric congestion games