Compatibility of Fairness and Nash Welfare under Subadditive Valuations
Siddharth Barman, Mashbat Suzuki
TL;DR
This work studies the compatibility of fairness and efficiency in allocating indivisible goods under subadditive valuations. It develops existential and computational results showing that EF1 and partial EFx can be achieved with a constant-factor loss in Nash social welfare (NSW), specifically NSW ≥ 1/2 NSW* for EF1 and NSW ≥ 1/2 NSW* for partial EFx, while a polynomial-time procedure yields EF1 with NSW at least NSW(input)/2.08, establishing O(1)-approximation guarantees and improving prior O(n) bounds in EF1 settings. The results extend to constrained down-sets and asymmetric NSW, providing a versatile black-box method to convert any efficient outcome into a fairly balanced one with only a modest decrease in social welfare. The findings significantly broaden the landscape of fair division under broad valuation classes and offer practical algorithmic tools for achieving fair and efficient allocations.
Abstract
We establish a compatibility between fairness and efficiency, captured via Nash Social Welfare (NSW), under the broad class of subadditive valuations. We prove that, for subadditive valuations, there always exists a partial allocation that is envy-free up to the removal of any good (EFx) and has NSW at least half of the optimal; here, optimality is considered across all allocations, fair or otherwise. We also prove, for subadditive valuations, the universal existence of complete allocations that are envy-free up to one good (EF1) and also achieve a factor $1/2$ approximation to the optimal NSW. Our EF1 result resolves an open question posed by Garg, Husic, Li, Végh, and Vondrák (STOC 2023). In addition, we develop a polynomial-time algorithm which, given an arbitrary allocation $\widetilde{A}$ as input, returns an EF1 allocation with NSW at least $\frac{1}{e^{2/e}}\approx \frac{1}{2.08}$ times that of $\widetilde{A}$. Therefore, our results imply that the EF1 criterion can be attained simultaneously with a constant-factor approximation to optimal NSW in polynomial time (with demand queries), for subadditive valuations. The previously best-known approximation factor for optimal NSW, under EF1 and among $n$ agents, was $O(n)$ -- we improve this bound to $O(1)$. It is known that EF1 and exact Pareto efficiency (PO) are incompatible with subadditive valuations. Complementary to this negative result, the current work shows that we regain compatibility by just considering a factor $1/2$ approximation: EF1 can be achieved in conjunction with $\frac{1}{2}$-PO under subadditive valuations. As such, our results serve as a general tool that can be used as a black box to convert any efficient outcome into a fair one, with only a marginal decrease in efficiency.