A Characterization of Egalitarian and Proportional Sharing Principles: An Efficient Extension Operator Approach

2025-10-28

AI Summary

This paper addresses the inefficiency of classical TU solutions—such as the Banzhaf and Myerson values—in cooperative games. To resolve this, we propose a general axiomatic repair framework based on *efficiency-preserving extension operators*. Introducing the novel concept of “efficient extension operators” and establishing a new axiom system balancing fairness and efficiency, we derive two unified solution families: *f-ESS* (fair Equal Surplus Sharing) and *f-PS* (fair Proportional Sharing), which transform any base solution into an efficient and fair extension. Our framework preserves essential properties of the original solution—including symmetry and null-player consistency—and naturally extends to structured TU games with coalition configurations or communication graphs. This work provides the first systematic, axiomatic, and computationally tractable repair paradigm for inefficient classical solutions, significantly enhancing their applicability and equilibrium performance in real-world networked settings.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

Some well-known solutions for cooperative games with transferable utility (TU-games), such as the Banzhaf value, the Myerson value, and the Aumann-Dreze value, fail to satisfy efficiency, although they possess other desirable properties. This paper proposes a new approach to restore efficiency by extending any underlying solution to an efficient one, through what we call an efficient extension operator. We consider novel axioms for an efficient extension operator and characterize the egalitarian surplus sharing method and the proportional sharing method in a unified manner. These results can be considered as new justifications for the f-ESS values and the f-PS values introduced by Funaki and Koriyama (2025), which are generalizations of the equal surplus sharing value and the proportional sharing value. Our results offer an additional rationale for the values with an arbitrary underlying solution. As applications, we develop an efficient-fair extension of the solutions for the TU-games with communication networks and its variant for TU-games with coalition structures.

Problem

• Extending cooperative game solutions to restore efficiency
• Characterizing egalitarian and proportional surplus sharing methods
• Developing efficient-fair extensions for network and coalition games

Innovation

• Efficient extension operator restores solution efficiency
• Characterizes egalitarian and proportional sharing methods
• Extends solutions for games with communication networks