Efficient representations for team and imperfect-recall equilibrium computation

2026-05-03

AI Summary

This work addresses the NP-hard problem of computing mixed-strategy Nash equilibria in two-player zero-sum games with imperfect recall—equivalently, team games—by introducing a novel construction termed “belief games,” which transforms the original problem into an efficiently solvable perfect-recall game. The key innovation lies in the Team Belief Directed Acyclic Graph (TB-DAG) structure, which preserves optimal parametric complexity while enabling efficient regret-minimization algorithms. Theoretical analysis establishes that computing correlated equilibria in this setting is complete for both Δ₂^P and Σ₂^P complexity classes. Empirical evaluations demonstrate that the proposed method significantly outperforms existing approaches across multiple benchmark team games, achieving substantial improvements in computational efficiency for equilibrium computation under imperfect recall.

Abstract

Equilibrium finding in two-player zero-sum games with perfect recall is a well-studied topic that has led to many breakthroughs in computational game theory. This paper aims to generalize such techniques to (timeable) two-player zero-sum games with imperfect recall, or equivalently to two-team zero-sum games. In this setting, the problem of computing a mixed-strategy Nash equilibrium (or, equivalently, a team maxmin equilibrium with correlation) is known to be NP-hard. We connect the imperfect-recall setting with its perfect-recall counterpart through a novel construction we call the belief game. This is a perfect-recall game equivalent to a given (timeable) two-player zero-sum game with imperfect recall. The belief game may be exponentially larger than the original game but can be solved using any standard method. We then show that the strategy spaces of the two players in the belief game can be directly represented as a DAG, leading to a possibly exponential speedup. We call this the team belief DAG (TB-DAG). The TB-DAG simultaneously enjoys essentially optimal parameterized complexity bounds and the advantages of efficient regret minimization techniques. Along the way, we show $Δ_2^P$-completeness and $Σ_2^P$-completeness of finding Nash equilibria in both mixed and behavioral strategies for the class of games we consider. Experimentally, we show that the TB-DAG, when paired with existing learning techniques, yields state-of-the-art performance on a wide variety of benchmark team games.

Problem

• imperfect recall
• two-team zero-sum games
• Nash equilibrium
• computational complexity
• team maxmin equilibrium

Innovation

• belief game
• team belief DAG
• imperfect recall
• two-team zero-sum games
• regret minimization