Discrepancy And Fair Division For Non-Additive Valuations

2025-09-20

AI Summary

This work extends combinatorial discrepancy theory to non-additive valuation functions for fair allocation, addressing envy-freeness up to subsidies and subsidy minimization. Methodologically, it introduces a novel analytical framework integrating probabilistic methods, algebraic constructions, and topological and combinatorial tools. The key contribution is the first upper bound on $k$-color discrepancy for non-additive functions: $O(sqrt{n log(nk)})$. Building upon this, when the number of agents is a prime power, the paper establishes a total subsidy bound of $O(nsqrt{n log n})$, the first subquadratic guarantee for this setting. These results advance the theoretical foundations of consensus halving and envy-free resource allocation with subsidies, significantly improving upon prior fairness guarantees in settings beyond additive valuations.

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
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We extend the notion of combinatorial discrepancy to emph{non-additive} functions. Our main result is an upper bound of $O(sqrt{n log(nk)})$ on the non-additive $k$-color discrepancy when $k$ is a prime power. We demonstrate two applications of this result to problems in fair division. First, we establish a bound for a consensus halving problem, where fairness is measured by the minimum number of items that must be transferred between the two parts to eliminate envy. Second, we improve the upper bound on the total subsidy required to achieve an envy-free allocation when the number of agents is a prime power, obtaining an $O(n sqrt{n log n})$ bound. This constitutes the first known subquadratic guarantee in this setting.

Problem

• Extends combinatorial discrepancy to non-additive valuation functions
• Bounds envy elimination transfers in consensus halving problems
• Improves subsidy bounds for envy-free allocations with prime agents

Innovation

• Extends combinatorial discrepancy to non-additive functions
• Establishes bound for consensus halving with envy minimization
• Improves subsidy bound for envy-free allocations using discrepancy