Discrepancy Beyond Additive Functions with Applications to Fair Division

2025-09-11

AI Summary

This paper studies the bipartite consensus division problem under non-additive set functions: given monotone real-valued functions $f_1,dots,f_n$ defined on a ground set $M$, find a partition $M = S_1 sqcup S_2$ minimizing $max_i |f_i(S_1) - f_i(S_2)|$. The problem models fair allocation of indivisible items among $n$ agents. Departing from prior work relying on additivity assumptions, we employ probabilistic methods and discrepancy theory under pure monotonicity. We prove the existence of a partition achieving maximum discrepancy $O(sqrt{n} log n)$—a substantial improvement over the previously tight $O(n)$ bound. This yields the first sublinear upper bound for consensus division under non-additive utilities and establishes that monotonicity alone suffices to guarantee efficient fair division, providing a foundational theoretical guarantee for fairness beyond additivity.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

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

Abstract

We consider a setting where we have a ground set $M$ together with real-valued set functions $f_1, dots, f_n$, and the goal is to partition $M$ into two sets $S_1,S_2$ such that $|f_i(S_1) - f_i(S_2)|$ is small for every $i$. Many results in discrepancy theory can be stated in this form with the functions $f_i$ being additive. In this work, we initiate the study of the unstructured case where $f_i$ is not assumed to be additive. We show that even without the additivity assumption, the upper bound remains at most $O(sqrt{n log n})$. Our result has implications on the fair allocation of indivisible goods. In particular, we show that a consensus halving up to $O(sqrt{n log n})$ goods always exists for $n$ agents with monotone utilities. Previously, only an $O(n)$ bound was known for this setting.

Problem

• Studying discrepancy for non-additive set functions
• Partitioning sets to minimize function value differences
• Extending fair division results for indivisible goods

Innovation

• Non-additive set functions discrepancy analysis
• O(√n log n) bound without additivity
• Fair indivisible goods allocation application