Union Shapley Value: Quantifying Group Impact via Collective Removal

2025-05-27

AI Summary

Existing group-level attribution methods lack a rigorous axiomatic extension of the Shapley value. Method: We propose the Union Shapley Value (USV), a novel measure that quantifies a group’s marginal contribution in cooperative games by removing the entire group at once—formalizing the intuitive notion of “group deletion” for the first time. Contribution/Results: We establish two complete axiomatizations proving USV’s uniqueness and reveal its intrinsic connections to classical group values such as the Banzhaf and Owen values. Furthermore, we introduce a dual synergy framework that unifies multiple existing group attribution methods. Theoretically, USV is the unique solution satisfying efficiency, group symmetry, and group marginal contribution additivity—rendering it both interpretable and computationally feasible. Our work thus provides a rigorous axiomatic foundation and a practical tool for group-level attribution.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Humans and AI: Learning Human Values and Preferences
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Extends Shapley value to evaluate group contributions collectively
• Proposes Union Shapley Value by analyzing group removal impact
• Explores axiomatic links between group values and synergy measures

Innovation

• Proposes Union Shapley Value for group impact
• Uses axiomatic characterizations for formalization
• Identifies dual approach measuring synergy