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.
翻译:尽管在计算游戏理论方面最近取得了许多实际和理论上的突破,但在广泛组合团队游戏中找到平衡仍然是一项重大挑战。虽然在最坏的情况下,NP-hard在最坏的情况下是硬的,但对于团队游戏中的某些家庭来说,却有可以想象的高效算法。特别是,如果游戏有共同的外部信息,也称为A损失回顾 -- -- 非正式地,非团队成员(即对立团队或自然)的行动不是整个团队所不知道的,就是团队内部的共同知识 -- -- 然后多盘点算法(Kaneko和Kline,1995年)。在本文中,我们设计了一个全新的游戏游戏逻辑。我们设计了一个全新的算法,用来解决团队游戏游戏游戏游戏的游戏。它使用树形的制约系统来减少正确性需要的数量和程度(数学程序的紧缩),我们的算法把团队游戏的线性程序分为最多以NW+w+O+O+1美元计价, 在游戏的首个分数中, 双价算为游戏树的大小, 美元是直接的参数, 在游戏游戏游戏机队的直径的直径直径直径直径直径上, 显示我们的游戏机型游戏的直径直径直径直径的游戏的游戏。