Asymptotic Fair Division: Chores Are Easier Than Goods

When dividing items among agents, two of the most widely studied fairness notions are envy-freeness and proportionality. We consider a setting where m chores are allocated to n agents and the disutility of each chore for each agent is drawn from a probability distribution. We show that an envy-free allocation exists with high probability provided that m ≥ 2n, and moreover, m must be at least n + Θ(n) in order for the existence to hold.
On the other hand, we prove that a proportional allocation is likely to exist as long as m = ω(1), and this threshold is asymptotically tight. Our results reveal a clear contrast with the allocation of goods, where a larger number of items is necessary to ensure existence for both notions.