Despite the many recent practical and theoretical breakthroughs in computational game theory, equilibrium finding in extensive-form team games remains a significant challenge. While NP-hard in the worst case, there are provably efficient algorithms for certain families of team game. In particular, if the game has common external information, also known as A-loss recall -- informally, actions played by non-team members (i.e., the opposing team or nature) are either unknown to the entire team, or common knowledge within the team -- then polynomial-time algorithms exist (Kaneko and Kline, 1995). In this paper, we devise a completely new algorithm for solving team games. It uses a tree decomposition of the constraint system representing each team's strategy to reduce the number and degree of constraints required for correctness (tightness of the mathematical program). Our algorithm reduces the problem of solving team games to a linear program with at most $NW^{w+O(1)}$ nonzero entries in the constraint matrix, where $N$ is the size of the game tree, $w$ is a parameter that depends on the amount of uncommon external information, and $W$ is the treewidth of the tree decomposition. In public-action games, our program size is bounded by the tighter $\tilde O(3^t 2^{t(n-1)}NW)$ for teams of $n$ players with $t$ types each. Since our algorithm describes the polytope of correlated strategies directly, we get equilibrium finding in correlated strategies for free -- instead of, say, having to run a double oracle algorithm. We show via experiments on a standard suite of games that our algorithm achieves state-of-the-art performance on all benchmark game classes except one. We also present, to our knowledge, the first experiments for this setting where more than one team has more than one member.

中文翻译：

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通过树分解的零和扩展形式博弈中的团队相关均衡

尽管最近在计算博弈论中取得了许多实践和理论突破，但在广泛形式的团队博弈中寻找均衡仍然是一个重大挑战。虽然在最坏的情况下是 NP-hard，但对于某些团队游戏系列，有可证明有效的算法。特别是，如果游戏有共同的外部信息，也称为 A-loss 召回——非正式地，非团队成员（即对方团队或性质）的行为要么是整个团队都不知道的，要么是内部的共同知识团队——然后多项式时间算法存在（Kaneko 和 Kline，1995）。在本文中，我们设计了一种全新的解决团队游戏的算法。它使用代表每个团队的约束系统的树分解 减少正确性（数学程序的严密性）所需约束的数量和程度的策略。我们的算法将解决团队游戏的问题简化为线性程序，约束矩阵中最多包含 $NW^{w+O(1)}$ 个非零项，其中 $N$ 是博弈树的大小，$w$是一个取决于不常见的外部信息量的参数，$W$ 是树分解的树宽。在公共动作游戏中，我们的程序大小受到更严格的 $\tilde O(3^t 2^{t(n-1)}NW)$ 的限制，适用于每个 $n$ 类型的 $n$ 玩家的团队。由于我们的算法直接描述了相关策略的多面体，我们可以免费在相关策略中找到均衡——而不是，比如说，必须运行双预言机算法。我们通过对标准游戏套件的实验表明，我们的算法在除一个之外的所有基准游戏类上都达到了最先进的性能。据我们所知，我们还针对这种设置进行了第一次实验，其中多个团队拥有多个成员。