Recovering Weighted Shapley Values from Position Preferences

2026-10-07

AI Summary

This study addresses the problem of recovering weighted Shapley values from positional preferences. Building upon Roth’s framework, it interprets weighted Shapley values in transferable utility games as positional preferences. By integrating axiomatic economics, probability theory, and constructive proof techniques, this work proposes the probabilistic equivalence of positional values alongside separable Luce conditions, and establishes extended uniqueness for the two-player case. The primary contributions include a complete characterization of the Kalai–Samet family, the construction of a formal mapping between preferences and weights, and the verification of axiom independence. Collectively, these results offer a novel perspective on the axiomatic characterization of cooperative games.

Abstract

Weighted Shapley values represent asymmetry through positive weights and ordered priority classes. We give these parameters a preference interpretation in Roth's framework of lotteries over positions in transferable-utility games. Under common calibration, nullity, ordinary risk neutrality, unanimity efficiency, and unanimity positivity, position values in unanimity games are probability equivalents. Luce conditioning of these probabilities characterizes the Kalai-Samet family and is equivalent to partnership consistency. We separate conditioning into transitivity of revealed priority, pairwise determination of support, and invariance of positive odds. Two-player observations admit a weighted Shapley extension exactly when revealed priority is transitive and within-class odds satisfy a product rule. The extension is unique: pairwise comparisons identify the priority classes and weight ratios within each class. Larger unanimity games provide additional restrictions on supports and shares. We give constructive proofs, establish independence of the five axioms, and explain the scope and limits of the elicitation interpretation.

Problem

• Weighted Shapley values
• Position preferences
• Transferable-utility games
• Unanimity games
• Kalai-Samet family

Innovation

• Weighted Shapley Values
• Position Preferences
• Luce Conditioning
• Partnership Consistency
• Axiomatic Characterization