A Little Charity Guarantees Almost EnvyFreeness Fair division of indivisible goods is a very wellstudied problem. The goal of this problem is to distribute m goods to n agents in a fair manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envyfreeness is the most extensively studied notion of fairness. However, envyfree allocations do not always exist when goods are indivisible. The notion of fairness we consider here is envyfreeness up to any good (EFX) where no agent envies another agent after the removal of any single good from the other agentu0027s bundle. It is not known if such an allocation always exists even when n3. We show there is always a partition of the set of goods into n1 subsets (X_1,ldots,X_n,P) where for i in n, X_i is the bundle allocated to agent i and the set P is unallocated (or donated to charity) such that we havecolon 1) envyfreeness up to any good, 2) no agent values P higher than her own bundle, and 3) fewer than n goods go to charity, i.e., P u003c n (typically m gg n). Our proof is constructive. When agents have additive valuations and lvert P rvert is large (i.e., when P is close to n), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is 47 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 12 known for an approximate GMMS allocation.

A Simple and Approximately Optimal Mechanism for an Additive Buyer We consider a monopolist seller with n heterogeneousitems, facing a single buyer. The buyer hasa value for each item drawn independently according to(nonidentical) distributions, and his value for a set ofitems is additive. The seller aims to maximize his revenue.It is known that an optimal mechanism in this setting maybe quite complex, requiring randomization 19 and menusof infinite size 15. Hart and Nisan 17 have initiated astudy of two very simple pricing schemes for this setting:item pricing, in which each item is priced at its monopolyreserve; and bundle pricing, in which the entire set ofitems is priced and sold as one bundle. Hart and Nisan 17have shown that neither scheme can guarantee more thana vanishingly small fraction of the optimal revenue. Insharp contrast, we show that for any distributions, thebetter of item and bundle pricing is a constantfactorapproximation to the optimal revenue. We further discussextensions to multiple buyers and to valuations that arecorrelated across items.