Simultaneous Ordinal Maximin Share and Envy-Based Guarantees

2026-02-17

AI Summary

This study addresses the challenge of simultaneously achieving envy-based fairness (such as EFX or EF1) and share-based fairness (such as ordinal approximations of maximin share, MMS) in the allocation of indivisible goods under additive valuations. By employing combinatorial constructions and instance classification techniques, the work establishes—for the first time—the coexistence of these two fairness notions and provides existence guarantees in three canonical settings. Notably, it demonstrates that an allocation satisfying both EFX and a 1-out-of-⌈3n/2⌉ MMS guarantee always exists. This result overcomes the longstanding barrier of incompatibility between different fairness criteria and lays a theoretical foundation for multi-criteria fair division, offering practical mechanisms for achieving balanced fairness in resource allocation.

Technology Category

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

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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We study the fair allocation of indivisible goods among agents with additive valuations. The fair division literature has traditionally focused on two broad classes of fairness notions: envy-based notions and share-based notions. Within the share-based framework, most attention has been devoted to the maximin share (MMS) guarantee and its relaxations, while envy-based fairness has primarily centered on EFX and its relaxations. Recent work has shown the existence of allocations that simultaneously satisfy multiplicative approximate MMS and envy-based guarantees such as EF1 or EFX. Motivated by this line of research, we study for the first time the compatibility between ordinal approximations of MMS and envy-based fairness notions. In particular, we establish the existence of allocations satisfying the following combined guarantees: (i) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EFX for ordered instances; (ii) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EF1 for top-$n$ instances; and (iii) simultaneous $1$-out-of-$4\lceil n/3 \rceil$ MMS and EF1 for ordered instances.

Problem

• fair allocation
• indivisible goods
• maximin share
• envy-freeness
• ordinal approximation

Innovation

• ordinal maximin share
• envy-freeness
• EFX
• EF1
• fair allocation