Achieving Proportionality up to the Maximin Item with Indivisible Goods

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