Comparing Approximate Relaxations of Envy-Freeness

TL;DR

This paper explores the relationships and approximation guarantees among various relaxed fairness notions in indivisible goods division, revealing surprising equivalences and providing a comprehensive understanding of their comparative strengths.

Contribution

It establishes tight bounds and connections between four key fairness relaxations and their approximate versions, enhancing understanding of their relative power.

Findings

PMMS and EFX provide the same worst-case guarantee for MMS

Several tight or nearly tight bounds on approximation quality

Insights into the power and limitations of fairness notions

Abstract

In fair division problems with indivisible goods it is well known that one cannot have any guarantees for the classic fairness notions of envy-freeness and proportionality. As a result, several relaxations have been introduced, most of which in quite recent works. We focus on four such notions, namely envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin share fairness (MMS), and pairwise maximin share fairness (PMMS). Since obtaining these relaxations also turns out to be problematic in several scenarios, approximate versions of them have been considered. In this work, we investigate further the connections between the four notions mentioned above and their approximate versions. We establish several tight, or almost tight, results concerning the approximation quality that any of these notions guarantees for the others, providing an almost complete picture of this landscape. Some of our findings reveal interesting and surprising consequences regarding the power of these notions, e.g., PMMS and EFX provide the same worst-case guarantee for MMS, despite PMMS being a strictly stronger notion than EFX. We believe such implications provide further insight on the quality of approximately fair solutions.