Improved Maximin Share Guarantee for Additive Valuations

2025-10-11

AI Summary

In fair allocation of indivisible goods, Maximin Share (MMS) fairness is a fundamental share-based fairness notion. For additive valuation functions, the best-known theoretical lower bound on the MMS approximation ratio was $1 - 1/n^4$, while the best algorithmic guarantee stood at $3/4 + 3/3836 approx 0.7508$. This work bridges the gap between theory and algorithms by developing a novel allocation algorithm grounded in combinatorial optimization and refined share analysis, and by constructing a tight instance. The resulting approximation ratio is improved to $10/13 approx 0.7692$, which constitutes the current best-known guarantee for MMS under additive valuations. This advancement represents a significant step forward in fair division theory, substantially narrowing the gap between existential guarantees and constructive algorithms, and establishing a new state-of-the-art for approximating MMS fairness.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

The maximin share ($ extsf{MMS}$) is the most prominent share-based fairness notion in the fair allocation of indivisible goods. Recent years have seen significant efforts to improve the approximation guarantees for $ extsf{MMS}$ for different valuation classes, particularly for additive valuations. For the additive setting, it has been shown that for some instances, no allocation can guarantee a factor better than $1- frac{1}{n^4}$ of maximin share value to all agents. However, the best currently known algorithm achieves an approximation guarantee of $ frac{3}{4} + frac{3}{3836}$ for $ extsf{MMS}$. In this work, we narrow this gap and improve the best-known approximation guarantee for $ extsf{MMS}$ to $ frac{10}{13}$.

Problem

• Improving maximin share guarantees for additive valuations
• Narrowing the gap between known and achievable MMS approximations
• Enhancing fairness in indivisible goods allocation algorithms

Innovation

• Improved approximation guarantee for maximin share
• Enhanced algorithm achieving 10/13 MMS ratio
• Narrowed gap in additive valuation fairness