Envy-Free and Pareto-Optimal Allocations for Agents with Asymmetric Random Valuations

TL;DR

This paper investigates the existence of envy-free and Pareto-optimal allocations for indivisible items among agents with asymmetric, agent-specific utility distributions, providing probabilistic guarantees and a polynomial-time algorithm.

Contribution

It generalizes previous symmetric models to asymmetric distributions, proving existence of fair and efficient allocations under broader conditions and offering an efficient algorithm.

Findings

Envy-free and Pareto-optimal allocations likely exist when items are proportional to n log n.

The existence guarantees hold with high probability under the asymmetric model.

A polynomial-time algorithm can find such allocations when conditions are met.

Abstract

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 where each agent's utilities 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 $m=\Omega\left(n\log n\right)$, which is tight up to a log log gap that also remains open in the symmetric subsetting. Furthermore, these guarantees can be achieved by a polynomial-time algorithm.