The Unreasonable Fairness of Maximum Nash Welfare

Abstract

The maximum Nash welfare (MNW) solution -which selects an allocation that maximizes the product of utilities -is known to provide outstanding fairness guarantees when allocating divisible goods. And while it seems to lose its luster when applied to indivisible goods, we show that, in fact, the MNW solution is unexpectedly, strikingly fair even in that setting. In particular, we prove that it selects allocations that are envy free up to one good -a compelling notion that is quite elusive when coupled with economic efficiency. We also establish that the MNW solution provides a good approximation to another popular (yet possibly infeasible) fairness property, the maximin share guarantee, in theory and -even more so -in practice. While finding the MNW solution is computationally hard, we develop a nontrivial implementation, and demonstrate that it scales well on real data. These results lead us to believe that MNW is the ultimate solution for allocating indivisible goods, and underlie its deployment on a popular fair division website.

Nash Social Welfare for Indivisible Items under Separable, Piecewise-Linear Concave Utilities

“…NSW is scale free and it achieves a natural compromise between fairness and efficiency, see [20,6] also for other properties of NSW. Additionally, NSW is related, directly or indirectly, to several key solution concepts in game theory and economics.…”

Almost Envy-Freeness with General Valuations

“…These mechanisms are available for public use at no cost: users can simply log in, define what is to be divided, and enter their valuations. Since the site's launch in November 2014, there have been over 60,000 users [10]. The company Fair Outcomes, Inc. (http://www.fairoutcomes.com) offers fair division services in a similar vein.…”

“…For example, in the case of distribution of indivisible goods on Spliddit, users know that the solution will be envy-free up to one good and Pareto optimal [10]. Our hope is that further work in the area of fair division of indivisible goods will allow user-facing services like Spliddit, Fair Outcomes, Inc., and Course Match to offer users even stronger fairness guarantees.…”

“…In contrast, the algorithm of Lipton et al [20] does not guarantee a Pareto optimal allocation. 2 Caragiannis et al [10] also proposed the fairness criterion of envy-freeness up to any good, and left the possible existence of EFX allocations as an open problem. We are not aware of any results regarding EFX allocations prior to this work.…”

Fair and Efficient Social Choice in Dynamic Settings

“…Caragiannis et al [2016] have recently shown the MNW solution to satisfy envy freeness up to one good, as well as approximating the maximin share guarantee. However, work in fair division focuses primarily on the allocation of private goods, where each alternative gives utility to exactly one agent.…”