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 et al. (STOC 2023). In addition, we develop a polynomial-time algorithm which, given an arbitrary allocation \~A as input, returns an EF1 allocation with NSW at least $1/3$ times that of \~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.

翻译：我们在次可加估值这一广泛类别下，建立了公平性与效率（通过纳什社会福利（NSW）衡量）之间的兼容性。我们证明：对于次可加估值，总存在一个部分分配，它在移除任意物品后是无嫉妒的（EFx），且其NSW至少达到最优值的一半；此处的最优性是在所有分配（无论公平与否）中考虑的。我们还证明：对于次可加估值，始终存在完整的分配方案，满足移除一个物品后无嫉妒（EF1），同时能达到最优NSW的$1/2$近似。我们的EF1结果解决了Garg等人（STOC 2023）提出的一个开放性问题。此外，我们开发了一种多项式时间算法，该算法以任意分配\~A作为输入，返回一个EF1分配，其NSW至少为\~A的$1/3$倍。因此，我们的结果表明，对于次可加估值，EF1准则可以与最优NSW的常数因子近似在多项式时间内（通过需求查询）同时实现。此前在EF1约束下且针对$n$个智能体的最优NSW最佳已知近似因子为$O(n)$——我们将该界限改进至$O(1)$。已知EF1与精确帕累托效率（PO）在次可加估值下是不兼容的。作为这一否定结果的补充，当前研究表明，仅考虑$1/2$近似因子即可恢复兼容性：在次可加估值下，EF1可以与$\frac{1}{2}$-PO同时实现。因此，我们的结果可作为一种通用工具，作为黑箱将任何高效结果转化为公平结果，且仅伴随效率的边际下降。