Two straightforward methods to extend an assessment of individual elements to groups are to sum individual assessments or to treat the group as a single merged element and assess it accordingly. In this work, we analyze another natural approach based on sequential elimination: elements of the group are removed one by one, and their assessments are aggregated. We study this approach in the context of coalitional games and show that, for almost all semivalues, it does not depend on the order of players. In particular, we introduce a new group value, called the Union Shapley Value, and investigate its axiomatic properties. Our results build on a comprehensive analysis of group values in coalitional games. Specifically, we define a class of group (weak consistent) semivalues - a variant of semivalues satisfying a weak form of monotonicity. This framework allows us to clarify the differences between existing notions in the literature. We show that existing group values either assess the total worth of a group or measure its synergy. We distinguish these two approaches axiomatically and uncover a connection between the corresponding values. In particular, we show that the well-known Interaction Index is a synergistic counterpart of the value introduced by Marichal et al., which corresponds to the merge approach. The analysis also yields new synergistic group values associated with the Union Shapley Value, which we call the Intersection Shapley Value.Our results demonstrate that the sequential extension - and the Union Shapley value in particular - constitute one of the most natural extensions of player values to groups in coalitional games.

翻译：将个体评估扩展到群体的两种直接方法是：累加个体评估，或将群体视为单一合并个体进行评估。本文分析了另一种基于序贯消去（sequential elimination）的自然方法：逐个移除群体中的元素，并聚合其评估值。我们在联盟博弈（coalitional games）的背景下研究该方法，并证明：对于几乎所有半值（semivalues），该方法不依赖于玩家的顺序。特别地，我们引入了一种新的群体值——联盟夏普利值（Union Shapley Value），并探讨其公理性质。我们的研究成果建立在对联盟博弈中群体值的全面分析之上。具体而言，我们定义了一类群体（弱一致）半值——满足弱单调性形式的半值变体。该框架使我们能够厘清文献中现有概念之间的差异。我们证明，既有群体值要么评估群体的总价值，要么衡量其协同效应。我们通过公理区分了这两种方法，并揭示了相应值之间的联系。特别地，我们证明著名的交互指数（Interaction Index）是Marichal等人所引入值（对应于合并方法）的协同对应物。该分析还产生了与联盟夏普利值相关的新的协同群体值——我们称之为交集夏普利值（Intersection Shapley Value）。我们的结果表明，序贯扩展方法——特别是联盟夏普利值——是联盟博弈中将玩家值扩展到群体最自然的方法之一。