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：将个体评估扩展至群体的两种直接方法包括：求和个体评估，或将群体视为单一合并元素并进行评估。本文基于序贯消除分析另一种自然方法：逐个移除群体元素并聚合其评估。我们在联盟博弈框架下研究该方法，证明对几乎所有半值而言，该结果不依赖玩家顺序。特别地，我们引入一种新的群体值——并集夏普利值，并探究其公理化性质。本文结果建立在对联盟博弈中群体值的系统分析之上：具体而言，我们定义了群体（弱一致性）半值类——满足弱单调性条件的半值变体。该框架厘清了现有文献中不同概念的差异。我们指出现有群体值要么评估群体总价值，要么衡量其协同效应。通过公理化方法区分这两种路径，揭示了对应值之间的内在联系：著名的交互指数正是Marichal等人提出的合并法对应值的协同效应对应物。分析还衍生出与并集夏普利值相关的新型协同群体值——我们称之为交集夏普利值。结果表明，序贯扩展方法（特别是并集夏普利值）是联盟博弈中将玩家值扩展至群体最自然的途径之一。