Solving Imperfect-Recall Games via Sum-of-Squares Optimization

2026-02-25

AI Summary

This work addresses the failure of conventional equilibrium computation methods in imperfect-recall extensive-form games due to information loss, proposing the first behavior-strategy-based Sum-of-Squares (SOS) optimization hierarchy. By leveraging the structure of information sets, the framework efficiently computes optimal strategies for single-player settings and Nash equilibria in multi-player games via semidefinite programming (SDP). The key contributions include the introduction of new classes of games—namely, (SOS)-concave and (SOS)-monotone games—and theoretical guarantees that the SOS hierarchy converges asymptotically in general cases and finitely in non-forgetful single-player games. For these newly defined game classes, a single SDP suffices to compute equilibria exactly, substantially enhancing both computational efficiency and applicability.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Search and Optimization: Adversarial Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Extensive-form games (EFGs) provide a powerful framework for modeling sequential decision making, capturing strategic interaction under imperfect information, chance events, and temporal structure. Most positive algorithmic and theoretical results for EFGs assume perfect recall, where players remember all past information and actions. We study the increasingly relevant setting of imperfect-recall EFGs (IREFGs), where players may forget parts of their history or previously acquired information, and where equilibrium computation is provably hard. We propose sum-of-squares (SOS) hierarchies for computing ex-ante optimal strategies in single-player IREFGs and Nash equilibria in multi-player IREFGs, working over behavioral strategies. Our theoretical results show that (i) these hierarchies converge asymptotically, (ii) under genericity assumptions, the convergence is finite, and (iii) in single-player non-absentminded IREFGs, convergence occurs at a finite level determined by the number of information sets. Finally, we introduce the new classes of (SOS)-concave and (SOS)-monotone IREFGs, and show that in the single-player setting the SOS hierarchy converges at the first level, enabling equilibrium computation with a single semidefinite program (SDP).

Problem

• Imperfect-recall games
• Extensive-form games
• Equilibrium computation
• Behavioral strategies
• Nash equilibrium

Innovation

• sum-of-squares optimization
• imperfect-recall games
• extensive-form games
• behavioral strategies
• semidefinite programming