Union Shapley Value: Quantifying Group Impact via Collective Removal
Piotr Kępczyński, Oskar Skibski
TL;DR
The paper tackles the challenge of quantifying group contributions in coalitional games beyond singleton Shapley values by introducing the Union Shapley value (US_S), defined via $US_S(N,v) = P(N,v) - P(N \setminus S, v)$ with $P(N,v) = ∑_{T ⊆ N, T ≠ ∅} Δ_v(T)/|T|$ and equivalently $US_S(N,v) = ∑_{T ⊆ N, S ∩ T ≠ ∅} Δ_v(T)/|T|$. It provides two axiomatic characterizations—Potential and Balanced Contributions—establishing US_S as the unique group value consistent with Shapley-value foundations at the group level, and connects this to a broader semivalue framework. The authors then characterize a class of group semivalues via nonnegative weights $p^q_t$ such that φ_S(N,v) = ∑_{T: S ∩ T ≠ ∅} p^{|S ∩ T|}_{|T|} Δ_v(T)$, showing US with $p^q_t = 1/t$ and related values like Merge Shapley and the sum of singleton Shapley values as special cases. A dual, synergistic perspective is developed through synergistic semivalues and the Intersection Shapley value, revealing deep connections (including duality and inclusion–exclusion relations) to the Union Shapley value and existing indices like the Interaction Index and Merge Shapley, with implications for applications to feature groups in machine learning and beyond.
Abstract
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.