The residual maximin share

2025-05-26

AI Summary

This paper studies fair allocation of indivisible goods among multiple agents with general monotone utility functions. We introduce a novel fairness benchmark—the Residual Maximin Share (RMMS)—the first formal definition of which ensures that RMMS is at least as large as the Maximin Share (MXS) and guarantees at least a 2/3-approximation to the Maximin Share (MMS) for additive utilities; it is both feasible and self-maximizing. Leveraging share theory and combinatorial existence arguments, we unify and strengthen classical fairness guarantees: we prove that an RMMS-and-EFX partial allocation always exists, and an RMMS-and-EFL complete allocation always exists. Our results extend the theoretical frontier of fair division and establish a new paradigm for strong fairness guarantees under general monotone utilities.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We consider fair allocations of indivisible goods to agents with general monotone valuations. We observe that it is useful to introduce a new share-based fairness notion, the {em residual maximin share} (RMMS). This share is {em feasible} and {em self maximizing}. Its value is at least as large as the MXS, and at least as large as $frac{2}{3}$-MMS for additive valuations. Known techniques easily imply the existence of partial allocations that are both RMMS and EFX, and complete allocations that are both RMMS and EFL. This unifies and somewhat improves upon several different results from previous papers.

Problem

• Fair allocation of indivisible goods among agents
• Introducing residual maximin share (RMMS) for fairness
• Ensuring RMMS feasibility and self-maximizing properties

Innovation

• Introduces residual maximin share (RMMS)
• Ensures RMMS is feasible and self-maximizing
• Unifies RMMS with EFX and EFL allocations