A Little Charity Guarantees Almost Envy-Freeness

Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good" (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when n=3.
We show there is always a partition of the set of goods into n+1 subsets (X_1,…,X_n,P) where for i∈[n], X_i is the bundle allocated to agent i and the set P is unallocated (or donated to charity) such that we have:

1) envy-freeness 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|<n (typically m≫n).

Our proof is constructive. When agents have additive valuations we show that we can obtain guarantees for many other notions of fairness simultaneously.