Achieving Proportionality up to the Maximin Item with Indivisible Goods
This work addresses fairness in resource allocation for multi-agent systems, offering a novel relaxation that bridges gaps between known approximations, though it is incremental in the context of existing research on proportionality relaxations.
The paper tackles the problem of fairly allocating indivisible goods by introducing a new fairness notion called proportionality up to the maximin item (PROPm), which serves as a middle-ground between existing approximations, and shows that it can be achieved for instances with up to five agents and additive valuations.
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.