Asymptotic Fair Division: Chores Are Easier Than Goods
Pasin Manurangsi, Warut Suksompong
TL;DR
This work analyzes the asymptotic existence of fair allocations in a chore-taking setting, where $m$ chores are assigned to $n$ agents and each agent $i$’s disutility for chore $j$ is drawn independently from a non-atomic distribution $ ons{D}$ on $[0,1]$ (often $(\alpha,\beta)$-PDF-bounded). The authors establish that envy-free allocations exist whp when $m \ge 2n$ (with a polynomial-time algorithm), while a lower bound shows nonexistence whp when $m \le (\nu-\epsilon)n$ with $\nu \approx 1.1256$; in contrast, proportional allocations exist whp for any $m=\omega(1)$, with a corresponding polynomial-time algorithm and matching lower bounds for constant $m$. The results rely on concentration inequalities, Efron–Stein bounds, and sophisticated matching arguments in random bipartite graphs, including two-stage constructions to handle indivisible $m$ and divisibility gaps. Overall, the paper reveals that achieving fairness is easier for chores than goods in the asymptotic sense, and it provides tight thresholds and efficient algorithms that illuminate the structure of fair division under random disutilities.
Abstract
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 $m \ge 2n$, 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.