When Do Envy-Free Allocations Exist?

We consider a fair division setting in which m indivisible items are to be allocated among n agents, where the agents have additive utilities and the agents' utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when m=Ω(nlog n) but not when $m=n+o(n)$, and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition---instead, the transition is governed by the divisibility relation between m and n. On the one hand, if m is divisible by n, an envy-free allocation exists with high probability as long as m≥ 2n. On the other hand, if m is not “almost” divisible by n, an envy-free allocation is unlikely to exist even when m=Θ(nlog n/loglog n).