Asymptotic existence of proportionally fair allocations

• We consider the asymptotic existence of proportionally fair allocations of indivisible goods.
• Proportionally fair allocations exist with high probability if the number of goods is a multiple of the number of agents.
• Proportionally fair allocations exist with high probability if the number of goods grows asymptotically faster than the number of agents.

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.