Efficient representations for team and imperfect-recall equilibrium computation

TL;DR

This paper introduces the belief game and team belief DAG (TB-DAG) to efficiently compute equilibria in imperfect-recall two-player zero-sum games, achieving theoretical and practical improvements.

Contribution

It presents a novel belief game construction and the TB-DAG representation, enabling faster equilibrium computation in complex imperfect-recall games.

Findings

TB-DAG enables exponential speedup in equilibrium computation.

The approach achieves state-of-the-art performance on benchmark team games.

Theoretical complexity results classify the difficulty of equilibrium finding.

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 $\Delta_2^P$-completeness and $\Sigma_2^P$-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.