Asymptotic Existence of Proportionally Fair Allocations

TL;DR

This paper investigates the likelihood of proportionally fair allocations of indivisible goods existing in large random settings, showing they are highly probable under certain growth conditions of goods relative to agents.

Contribution

It establishes asymptotic conditions under which proportionally fair allocations exist with high probability in random utility models.

Findings

Proportionally fair allocations exist with high probability when goods are a multiple of agents.

Existence is also likely when the number of goods grows faster than the number of agents.

Results apply to additive utilities with independently drawn utilities for goods.

Abstract

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.