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, V\'{e}gh, and Vondr\'{a}k (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.

翻译：我们在广泛的次可加估值类别下，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方案，该方案在移除任意物品后无嫉妒（EFx），且其NSW至少达到最优值的一半；此处的"最优"是指所有分配方案（无论是否公平）中的最优值。我们还证明，对于次可加估值，普遍存在完整的分配方案，这些方案在移除一件物品后无嫉妒（EF1），并且能以$1/2$的近似比达到最优NSW。我们的EF1结果解决了Garg、Husic、Li、V\'{e}gh和Vondr\'{a}k（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$\widetilde{A}$作为输入，返回一个EF1分配方案，其NSW至少达到$\widetilde{A}$的$\frac{1}{e^{2/e}}\approx \frac{1}{2.08}$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条件下，对于$n$个智能体，最优NSW的最佳已知近似因子为$O(n)$——我们将此界限改进为$O(1)$。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，当前工作表明，我们仅需考虑$1/2$的近似因子即可重获兼容性：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，作为一个黑箱使用，将任何高效结果转化为公平结果，而效率仅边际下降。