Finding Fair and Efficient Allocations

We study the problem of allocating indivisible goods fairly and efficiently among a set of agents who have additive valuations for the goods. Here, 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 to be efficient if it satisfies Pareto efficiency. A notable result of Caragiannis et al. (2016) shows that—under additive valuations—there is no need to trade efficiency for fairness; specifically, an allocation that maximizes the Nash social welfare (NSW) objective simultaneously satisfies these two seemingly incompatible properties. However, this approach does not directly provide an efficient algorithm for finding fair and efficient allocations, since maximizing NSW is an NP-hard problem. In this paper, we bypass this barrier, and develop a pseudo-polynomial time algorithm for finding allocations which are EF1 and Pareto efficient; 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 Pareto efficient. The approach developed in the paper leads to a polynomial-time 1.45-approximation for maximizing the Nash social welfare objective. This improves upon the best known approximation ratio for this problem. Our results are based on constructing Fisher markets wherein specific equilibria are not only efficient, but also fair.