Simultaneous Ordinal Maximin Share and Envy-Based Guarantees

TL;DR

This paper explores the compatibility of ordinal approximations of maximin share and envy-based fairness notions in fair division, establishing the existence of allocations satisfying combined guarantees.

Contribution

It introduces the first results on simultaneous ordinal MMS and envy-based fairness guarantees, expanding the understanding of fair division solutions.

Findings

Existence of allocations with simultaneous 1-out-of-⎡3n/2⎤ MMS and EFX for ordered instances.

Existence of allocations with simultaneous 1-out-of-⎡3n/2⎤ MMS and EF1 for top-n instances.

Existence of allocations with simultaneous 1-out-of-4⎡n/3⎤ MMS and EF1 for ordered instances.

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.