Envy-Free and Pareto-Optimal Allocations for Asymmetric Agents

We study the problem of allocating m indivisible items to n agents with additive utilities. It is desirable for the allocation to be both fair and efficient, which we formalize through the notions of envy-freeness and Pareto-optimality. While envy-free and Pareto-optimal allocations may not exist for arbitrary utility profiles, previous work has shown that such allocations exist with high probability assuming that all agents' values for all items are independently drawn from a common distribution. In this paper, we consider a generalization of this model with asymmetric agents, where an agent's utilities for the items are drawn independently from a distribution specific to the agent. We show that envy-free and Pareto-optimal allocations are likely to exist in this asymmetric model when $m=\Omega\left(n\, \log n\right)$, matching the best bounds known for the symmetric subsetting. Empirically, an algorithm based on Maximum Nash Welfare obtains envy-free and Pareto-optimal allocations for small numbers of items.