  Fair division has long been an important problem in the economics literature.
In this note, we consider the existence of proportionally fair allocations of
indivisible goods, i.e., allocations of indivisible goods in which every agent
gets at least her proportionally fair share according to her own utility
function. We show that when utilities are additive and utilities for individual
goods are drawn independently at random from a distribution, proportionally
fair allocations exist with high probability if the number of goods is a
multiple of the number of agents or if the number of goods grows asymptotically
faster than the number of agents.
