On the Complexity of the Optimal Correlated Equilibria in Extensive-Form Games

TL;DR

This paper investigates the computational complexity of finding optimal correlated equilibria in extensive-form games, revealing surprising hardness results and providing a comprehensive classification across multiple equilibrium concepts.

Contribution

It establishes the PSPACE-hardness of the Threshold problem for NFCE and NP-completeness for various correlated equilibrium concepts in extensive-form games, offering the most complete complexity landscape to date.

Findings

Threshold problem for NFCE is PSPACE-hard in multiplayer extensive-form games.

Threshold problem for Nash equilibria is ER-complete in this setting.

Threshold problem for AFCE is NP-hard even in two-player games without chance nodes.

Abstract

A major open question in algorithmic game theory is whether normal-form correlated equilibria (NFCE) can be computed efficiently in succinct games such as extensive-form games [DFF+25,6PR24,FP23,HvS08,VSF08,PR08]. Motivated by this question, we study the associated Threshold problem: deciding whether there exists a correlated equilibrium whose value exceeds a given threshold. We prove that this problem is PSPACE-hard for NFCE in multiplayer extensive-form games with perfect recall, even for fixed thresholds. To contextualize this result, we also establish the complexity of the Threshold problem for Nash equilibria in this setting, showing it is ER-complete. These results uncover a surprising complexity reversal: while optimal correlated equilibria are computationally simpler than optimal Nash in normal-form games, the opposite holds in extensive-form games, where computing optimal correlated equilibria is provably harder. Building on this line of inquiry, we also address a related question by [VSF08], who introduced the notions of extensive-form correlated equilibrium (EFCE) and agent-form correlated equilibrium (AFCE). They asked how difficult the Threshold problem is for AFCE; we answer this question by proving that it is NP-hard, even in two-player games without chance nodes. Complementing our hardness results, we establish tight complexity classifications for the Threshold problem across several correlated equilibrium concepts - including EFCE, AFCE, normal-form coarse, extensive-form coarse, and agent-form coarse correlated equilibria. For each of these solution concepts in multiplayer stochastic extensive-form games with perfect recall, we prove NP-completeness by providing matching NP upper bounds to the previously known hardness results. Together, our results provide the most complete landscape to date for the complexity of optimal equilibrium computation in extensive-form games.