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.

翻译：在具有完美回忆的两人零和博弈中寻找均衡是一个研究深入的主题，已在计算博弈论中引发许多突破。本文旨在将此类技术推广至（可时序化的）具有不完美回忆的两人零和博弈，或等价地推广至两人团队零和博弈。在此设定下，计算混合策略纳什均衡（或等价地，含关联的团队最大最小均衡）的问题已知为NP难问题。我们通过一种称为信念博弈的新颖构造，将不完美回忆设定与其完美回忆对应体联系起来。信念博弈是一种完美回忆博弈，等价于给定的（可时序化）具有不完美回忆的两人零和博弈。信念博弈可能比原始博弈规模指数级更大，但可通过任意标准方法求解。我们随后证明，信念博弈中两位玩家的策略空间可直接表示为有向无环图（DAG），从而带来潜在指数级加速。我们称之为团队信念DAG（TB-DAG）。TB-DAG同时享有本质上最优的参数化复杂度边界以及高效遗憾最小化技术的优势。在此过程中，我们证明了在所考虑的博弈类别中，混合策略和行为策略下的纳什均衡寻找问题分别为Δ₂^P-完全和Σ₂^P-完全。实验表明，TB-DAG与现有学习技术结合时，可在多种基准团队博弈上实现最先进性能。