Fair Allocations for Smoothed Utilities

Abstract

When allocating indivisible items across agents, it is desirable for the allocation to be envy-free, which means that each agent prefers their own bundle over every other bundle. Even though envy-free allocations are not guaranteed to exist for worst-case utilities, they frequently exist in practice. To explain this phenomenon, prior work has shown that, if utilities are drawn from certain probability distributions, then envy-free allocations exist with high probability (as long as the number of items is sufficiently large relative to the number of agents). In this paper, we study a more general setting, a smoothed model of utilities, in which utility profiles are mainly worst-case, but are slightly perturbed at random to avoid brittle counter-examples. Specifically, we start from a worst-case profile of utilities and, with some small probability, increase an agent's value for an item by adding a random amount, where the probability of perturbation and the distribution of perturbations are parameters of the model. If the probability of such perturbations is sufficiently large relative to the number of agents and items, we show that envy-free allocations exist with high probability and can be found efficiently. This analysis is tight up to constant factors. We also give an efficient algorithm for finding allocations that are simultaneously proportional and Pareto-optimal, which succeeds with high probability in the smoothed model.