Sequential Elimination and Union Shapley Value for Group Assessment in Coalitional Games

Abstract

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.