Finding Fair and Efficient Allocations

TL;DR

This paper presents a polynomial-time algorithm for finding fair and efficient allocations of indivisible goods, achieving EF1 and PO, and provides a better approximation for Nash social welfare than previous methods.

Contribution

It introduces a pseudopolynomial algorithm for EF1 and PO allocations, improves the approximation ratio for Nash social welfare, and establishes stronger existence results for fractional PO.

Findings

Algorithm finds EF1 and PO allocations in polynomial time when valuations are bounded.

Provides a 1.45-approximation for Nash social welfare, better than previous 2-approximation.

Establishes the existence of EF1 and fractionally PO allocations.

Abstract

We study the problem of allocating a set of indivisible goods among a set of agents in a fair and efficient manner. An allocation is said to be fair if it is envy-free up to one good (EF1), which means that each agent prefers its own bundle over the bundle of any other agent up to the removal of one good. In addition, an allocation is deemed efficient if it satisfies Pareto optimality (PO). While each of these well-studied properties is easy to achieve separately, achieving them together is far from obvious. Recently, Caragiannis et al. (2016) established the surprising result that when agents have additive valuations for the goods, there always exists an allocation that simultaneously satisfies these two seemingly incompatible properties. Specifically, they showed that an allocation that maximizes the Nash social welfare (NSW) objective is both EF1 and PO. However, the problem of maximizing NSW is NP-hard. As a result, this approach does not provide an efficient algorithm for finding a fair and efficient allocation. In this paper, we bypass this barrier, and develop a pseudopolynomial time algorithm for finding allocations that are EF1 and PO; in particular, when the valuations are bounded, our algorithm finds such an allocation in polynomial time. Furthermore, we establish a stronger existence result compared to Caragiannis et al. (2016): For additive valuations, there always exists an allocation that is EF1 and fractionally PO. Another contribution of our work is to show that our algorithm provides a polynomial-time 1.45-approximation to the NSW objective. This improves upon the best known approximation ratio for this problem (namely, the 2-approximation algorithm of Cole et al. (2017)). Unlike many of the existing approaches, our algorithm is completely combinatorial.