Recovering Weighted Shapley Values from Position Preferences

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.