Maximin Shares with Lower Quotas

2026-02-09

AI Summary

This study addresses the problem of achieving maximin share (MMS) fairness in the allocation of indivisible items (or tasks) among agents, subject to both lower and upper quota constraints on the number of items each agent receives. It extends the classical MMS framework to a generalized setting with heterogeneous additive valuations and quota constraints, further generalizing to multi-category goods. Leveraging combinatorial optimization and approximation algorithms, the work presents polynomial-time algorithms that guarantee, in the single-category case, a $\frac{2n}{3n-1}$-approximation of MMS for goods and a $\frac{3n-1}{2n}$-approximation for chores. In the multi-category setting, the corresponding approximation ratios are $\frac{n}{2n-1}$ for goods and $\frac{2n-1}{n}$ for chores.

Technology Category

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

Application Category

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

Abstract

We study the fair division of indivisible items among $n$ agents with heterogeneous additive valuations, subject to lower and upper quotas on the number of items allocated to each agent. Such constraints are crucial in various applications, ranging from personnel assignments to computing resource distribution. This paper focuses on the fairness criterion known as maximin shares (MMS) and its approximations. Under arbitrary lower and upper quotas, we show that a $\left(\frac{2n}{3n-1}\right)$-MMS allocation of goods exists and can be computed in polynomial time, while we also present a polynomial-time algorithm for finding a $\left(\frac{3n-1}{2n}\right)$-MMS allocation of chores. Furthermore, we consider the generalized scenario where items are partitioned into multiple categories, each with its own lower and upper quotas. In this setting, our algorithm computes an $\left(\frac{n}{2n-1}\right)$-MMS allocation of goods or a $\left(\frac{2n-1}{n}\right)$-MMS allocation of chores in polynomial time. These results extend previous work on the cardinality constraints, i.e., the special case where only upper quotas are imposed.

Problem

• maximin shares
• fair division
• indivisible items
• quota constraints
• additive valuations

Innovation

• maximin shares
• fair division
• quota constraints
• indivisible items
• polynomial-time algorithm