The Complexity of Equilibrium Refinements in Potential Games

TL;DR

The paper comprehensively maps the computational landscape of equilibrium refinements in potential games, showing $PLS$-completeness for pure perfect and proper equilibria across normal-form and extensive-form representations, while revealing NP-hard and coNP-hard barriers for normal-form proper equilibria in polytope games. It identifies tractable subclasses, notably symmetric matroid and network congestion games, where perfect equilibria are computable in polynomial time, and connects strongly polynomial-time and perturbed optimization to symbolic methods yielding broad algorithmic reach. It also uncovers exponential gaps in best-response path lengths between perfect and Nash equilibria and demonstrates that geometric proximity to perfect equilibria for mixed strategies can require doubly exponential precision in some 3-player settings, while polymatrix potential games admit CLS via perturbed gradient dynamics. Together, these results delineate when refinement concepts can meaningfully improve welfare and how computation interacts with game representation, offering both theoretical insights and practical implications for learning dynamics in multi-agent systems.

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 $\mathsf{CLS}$.

Figures (8)

• Figure 1: A $2 \times 2$ identical-interest game in normal form. It has exactly two Nash equilibria: (R1, C1) and (R2, C2). Of the two, only (R1, C1) is a perfect equilibrium.
• Figure 2: A $3 \times 3$ identical-interest game in normal form devised by Myerson78:Refinements.
• Figure 3: A $2 \times 2$ identical-interest game in normal form. Any strategy profile in which R1 is played with probability $1$ is a Nash equilibrium. The only perfect equilibrium is $(\textsf{R1}, \textsf{C1})$.
• Figure 4: Vanilla gradient descent (left) on the game of \ref{['fig:randomGD']} versus gradient descent on the perturbed game with perturbation magnitude $\epsilon \vcentcolon= 0.01$ (right) under $4$ different initializations. The welfare-optimal point and unique perfect equilibrium $(\textsf{R1}, \textsf{C1})$ appears in the bottom-left corner. We see that gradient descent in the unperturbed game often converges to Nash equilibria with much lower welfare than the optimal one.
• Figure 5: A $2 \times 2$ identical-interest game in normal form with a unique perfect equilibrium, namely (R1, C1).

Theorems & Definitions (139)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10