While coalitional games with externalities model a variety of real-life scenarios of interest to computer science, they pose significant game-theoretic and computational challenges. Specifically, key game-theoretic solution concepts—and the Shapley value in particular—can be extended to games with externalities in multiple, often orthogonal, ways. As for the computational challenges, while there exist two concise representations for coalitional games with externalities—called embedded MC-Nets and weighted MC-Nets—they allow the polynomial-time computation of only two of the six existing direct extensions of the Shapley values to games with externalities. In this article, inspired by the literature on endogenous coalition formation protocols, we propose to represent games with externalities in a way that mimic an intuitive process in which coalitions might form. To this end, we utilize Partition Decision Trees—rooted directed trees, where non-leaf nodes are labelled with agents’ names, leaf nodes are labelled with payoff vectors, and edges indicate membership of agents in coalitions. Interestingly, despite their apparent differences, the representation based on partition decision trees can be considered a subclass of embedded MC-Nets and weighted MC-Nets. The key advantage of this new representation is that it allows the polynomial-time computation of five out of six direct extensions of the Shapley value to games with externalities. In other words, by focusing on narrower Partition Decision Trees instead of wider embedded or weighted MC-Nets, a user is guaranteed to compute most extensions of the Shapley value in polynomial time.

中文翻译：

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分区决策树：有效计算Shapley值的表示形式，扩展到具有外部性的游戏

尽管具有外部性的联合游戏对计算机科学感兴趣的各种现实场景进行建模，但它们却带来了巨大的游戏理论和计算挑战。具体来说，关键的游戏理论解决方案概念（尤其是Shapley值）可以扩展为具有多种外部性（通常是正交的）的游戏。对于计算方面的挑战，尽管存在两种具有外部性的联合博弈的简洁表示形式（称为嵌入式MC-Net和加权MC-Net），但它们仅允许将Shapley值的六个现有直接扩展中的两个进行多项式时间计算。具有外部性的游戏。在本文中，受关于内生联盟形成协议的文献的启发，我们建议以模仿外部性的游戏方式来模仿可能形成联盟的直观过程。为此，我们利用分区决策树-根目录有向树，其中非叶子节点标记有代理的名称，叶子节点标记有支付向量，并且边缘指示联盟中的代理成员。有趣的是，尽管它们之间存在明显差异，但基于分区决策树的表示形式仍可以视为嵌入式MC-Net和加权MC-Net的子类。这种新表示形式的主要优点是，它允许Shapley值的六个直接扩展中的五个直接扩展到具有外部性的游戏的多项式时间计算。换句话说，通过专注于较窄的分区决策树，而不是较宽的嵌入式或加权MC-Net，可以确保用户在多项式时间内计算Shapley值的大多数扩展。