Union Shapley Value: Quantifying Group Impact via Collective Removal

We perform a comprehensive analysis of extensions of the Shapley value to groups. We propose a new, natural extension called the Union Shapley Value, which assesses a group's contribution by examining the impact of its removal from the game. This intuition is formalized through two axiomatic characterizations, closely related to existing axiomatizations of the Shapley value. Furthermore, we characterize the class of group semivalues and identify a dual approach that measures synergy instead of the value of a coalition. Our analysis reveals a novel connection between several group values previously proposed in the literature.