Achieving Proportionality up to the Maximin Item with Indivisible Goods

TL;DR

This paper introduces PROPm, a new fairness criterion for indivisible goods, and demonstrates its attainability for up to five agents, bridging the gap between existing fairness notions.

Contribution

The paper proposes PROPm, a novel fairness measure, and provides algorithms to achieve it for small instances, advancing fair division theory for indivisible goods.

Findings

PROPm is achievable for up to five agents with additive valuations.

PROPm offers a middle ground between PROP1 and PROPx fairness notions.

The approach connects proportionality relaxations with the maximin share concept.

Abstract

We study the problem of fairly allocating indivisible goods and focus on the classic fairness notion of proportionality. The indivisibility of the goods is long known to pose highly non-trivial obstacles to achieving fairness, and a very vibrant line of research has aimed to circumvent them using appropriate notions of approximate fairness. Recent work has established that even approximate versions of proportionality (PROPx) may be impossible to achieve even for small instances, while the best known achievable approximations (PROP1) are much weaker. We introduce the notion of proportionality up to the maximin item (PROPm) and show how to reach an allocation satisfying this notion for any instance involving up to five agents with additive valuations. PROPm provides a well-motivated middle-ground between PROP1 and PROPx, while also capturing some elements of the well-studied maximin share (MMS) benchmark: another relaxation of proportionality that has attracted a lot of attention.