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

TL;DR

This paper introduces sum-of-squares hierarchies for computing strategies in imperfect-recall extensive-form games, providing theoretical convergence guarantees and efficient algorithms for certain game classes.

Contribution

It develops SOS-based methods for equilibrium computation in IREFGs, including convergence analysis and new game classes with simplified solutions.

Findings

SOS hierarchies converge asymptotically

Finite convergence under genericity assumptions

Single-level SOS hierarchy for certain game classes

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).