Fair Allocation with Money: What is Your Objective?

2025-06-23

AI Summary

This paper investigates envy-free allocation of indivisible goods via monetary transfers—specifically balanced payments, uniform pricing, and uniform subsidies—and systematically compares their optimization objectives, including total absolute payment, total cost, or total subsidy. Method: Leveraging game-theoretic and optimization-theoretic frameworks, the authors establish formal connections among fairness constraints and minimization objectives across the three models. Contribution/Results: The work provides the first tight theoretical bounds on optimal solutions under each mechanism. Notably, the minimum total absolute payment under balanced payments is at least as large as the minimum total under uniform pricing or uniform subsidies, yet no more than twice that amount. Additional bounds are derived for related objectives, yielding provable performance guarantees. These results furnish both a rigorous theoretical benchmark and actionable design principles for implementing envy-free mechanisms in practice.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

When allocating indivisible items, there are various ways to use monetary transfers for eliminating envy. Particularly, one can apply a balanced vector of transfer payments, or charge each agent a positive amount, or -- contrarily -- give each agent a positive amount as a ``subsidy''. In each model, one can aim to minimize the amount of payments used; this aim translates into different optimization objectives in each setting. This note compares the various models, and the relations between upper and lower bounds for these objectives.

Problem

• Compare models for fair allocation with money transfers
• Minimize payment amounts in different envy-free settings
• Analyze bounds for optimization objectives in allocation

Innovation

• Balanced transfer payments eliminate envy
• Charge agents to minimize payments
• Subsidize agents to optimize objectives