A Little Charity Guarantees Almost Envy-Freeness

The fair division of indivisible goods is a very well-studied problem. goal this problem to distribute $m$ $n$ agents in “fair” manner, where every agent has valuation for each subset goods. We assume monotone valuations. Envy-freeness the most extensively studied notion fairness. However, envy-free allocations do not always exist when are indivisible. fairness we consider here “envy-freeness up any good,” EFX, no envies another after removal single good from other agent's bundle. It known if such an allocation exists. show there partition set into $n+1$ subsets $(X_1,\ldots,X_n,P)$, $i \in [n]$, $X_i$ bundle allocated $i$ and $P$ unallocated (or donated charity) that have (1) envy-freeness good, (2) values higher than her own bundle, (3) fewer go charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof constructive leads pseudopolynomial time algorithm find allocation. When additive valuations $|{P}|$ large (i.e., $|P|$ close $n$), our also maximin share (MMS) guarantee. Moreover, minor variant shows existence 4/7 groupwise (GMMS): stronger MMS. This improves upon current best bound 1/2 approximate GMMS (Very recently independently, Amanatidis, Ntokos, Markakis [Theoret. Comput. Sci., 841 (2020), pp. 94--109], showed 4/7-GMMS allocation.)