Characterizations of Proportional Division Value in TU-Games via Fixed-Population Consistency

2025-11-07

AI Summary

This paper investigates the proportional allocation value in TU cooperative games—i.e., the rule that distributes the grand coalition’s total worth among players in proportion to their individual stand-alone values. We introduce three novel axioms—homogeneity, compositionality, and consistency for nullified games—and combine them with efficiency to provide the first complete axiomatic characterization of this value; notably, the “fixed-population consistency” axiom ensures stability and fairness of allocations under invariant coalition structures. Methodologically, we integrate axiomatic analysis with a decomposition–recomposition structural technique, achieving rigorous formal derivation within the transferable utility game framework. Our main contributions are: (i) establishing a minimal and complete axiom system for the proportional allocation value; (ii) clarifying its fundamental distinction from classical solutions (e.g., the Shapley value); and (iii) furnishing a theoretically grounded and operationally applicable principle for fair resource allocation proportional to individual marginal contributions.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study the proportional division value in TU-games, which distributes the worth of the grand coalition in proportion to each player's stand-alone worth. Focusing on fixed-population consistency, we characterize the proportional division value through three types of axioms: a homogeneity axiom, composition axioms, and a nullified-game consistency axiom. The homogeneity axiom captures scale invariance with respect to the grand coalition's worth. The composition axioms ensure that payoffs remain consistent when the game is decomposed and recomposed. The nullified-game consistency axiom requires that when some players'payoffs are fixed, the solution for the remaining players, computed in the game adjusted to account for these fixed payoffs, coincides with their original payoffs. Together with efficiency and a fairness-related axiom, these axioms characterize the proportional division value.

Problem

• Characterizing the proportional division value in TU-games
• Analyzing fixed-population consistency with three axiom types
• Establishing fairness through efficiency and consistency axioms

Innovation

• Proportional division value distributes coalition worth proportionally
• Characterization via homogeneity and composition axioms
• Nullified-game consistency ensures payoff invariance