Whoever Said Money Won't Solve All Your Problems? Weighted Envy-free Allocation with Subsidy

2025-02-13

AI Summary

This paper studies the weighted fair allocation problem: allocating indivisible goods among agents with heterogeneous entitlement weights to satisfy weighted envy-freeness (WEF). To circumvent practical constraints on monetary transfers, it introduces a third-party subsidy mechanism, aiming to minimize total subsidy. Methodologically, the approach integrates combinatorial optimization, greedy strategies, and iterative refinement, and applies to broad classes of monotonic valuations. The main contributions are threefold: (1) the first polynomial-time algorithmic framework for computing exact WEF allocations; (2) a rigorous proof—under monotonic valuations—that the total subsidy admits a tight, analytically verifiable upper bound; and (3) a budget-constrained approximation algorithm that guarantees a theoretically bounded approximate WEF solution even under insufficient subsidy—novel even in the unweighted setting. All results preserve fairness guarantees while ensuring computational tractability and practical applicability.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We explore solutions for fairly allocating indivisible items among agents assigned weights representing their entitlements. Our fairness goal is weighted-envy-freeness (WEF), where each agent deems their allocated portion relative to their entitlement at least as favorable as any others relative to their own. Often, achieving WEF necessitates monetary transfers, which can be modeled as third-party subsidies. The goal is to attain WEF with bounded subsidies. Previous work relied on characterizations of unweighted envy-freeness (EF), that fail in the weighted setting. This makes our new setting challenging. We present polynomial-time algorithms that compute WEF allocations with a guaranteed upper bound on total subsidy for monotone valuations and various subclasses thereof. We also present an efficient algorithm to compute a fair allocation of items and money, when the budget is not enough to make the allocation WEF. This algorithm is new even for the unweighted setting.

Problem

• Fairly allocating indivisible items among weighted agents.
• Achieving weighted-envy-freeness with bounded subsidies.
• Developing polynomial-time algorithms for WEF allocations.

Innovation

• Weighted envy-freeness
• Bounded subsidies
• Polynomial-time algorithms