Fairly Dividing Non-identical Random Items: Just Sample or Match

Abstract

We study the question of existence and fast computation of fair and efficient allocations of indivisible resources among agents with additive valuations. As such allocations may not exist for arbitrary instances, we ask if they exist for typical or random instances, meaning when the utility values of agents for the resources are drawn from certain distributions. In this paper, we extend the previously studied formal models of this problem to non-identical items. We assume that every item is associated with a distribution U_j, and every agent's utility value for the item is drawn independently from U_j. We show that envy-free fair and maximum social welfare efficient allocations exist with high probability in the asymptotic setting, meaning when the number of agents n and items m are large. Further, we show that when m = Ω(n log n), then by only sampling O(log m) or O((log m)^2) utility values per item instead of all the n, we can compute these allocations in Õ(m) time. Finally, we simulate our algorithms on randomly generated instances and show that even for small instances, we suffer small multiplicative losses in the fairness and efficiency guarantees and converge to fully optimal guarantees quickly.