Weighted Envy Freeness With Limited Subsidies

2024-11-19

arXiv.org

AI Summary

This paper studies fair allocation of indivisible goods in weighted settings, targeting weighted envy-freeness (WEF): each agent, after normalizing valuations by their weight, weakly prefers their own bundle to any other. When WEF is unattainable via pure allocations, we introduce bounded external subsidies. We establish, for the first time in weighted settings, a theoretical characterization linking WEF feasibility to the minimum required subsidy—overcoming the limitation that envy-freeness (EF) characterizations from unweighted settings do not generalize. For general, identical, and binary additive valuation classes, we design polynomial-time algorithms and derive tight per-agent subsidy upper bounds; these recover known optimal bounds in the equal-weight special case. Our results unify and extend both the theoretical foundations and algorithmic frontiers of fair division.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 other's relative to their own. In many cases, achieving WEF necessitates monetary transfers, which can be modeled as third-party subsidies. The goal is to attain WEF with bounded subsidies. Previous work in the unweighted setting of subsidies relied on basic characterizations of EF that fail in the weighted settings. This makes our new setting challenging and theoretically intriguing. We present polynomial-time algorithms that compute WEF-able allocations with an upper bound on the subsidy per agent in three distinct additive valuation scenarios: (1) general, (2) identical, and (3) binary. When all weights are equal, our bounds reduce to the bounds derived in the literature for the unweighted setting.

Problem

• Allocating indivisible items fairly among weighted agents
• Achieving weighted envy-freeness with limited subsidies
• Developing polynomial-time algorithms for WEF allocations

Innovation

• Weighted envy-freeness fairness
• Polynomial-time allocation algorithms
• Bounded subsidies per agent