Compatibility of Fairness and Nash Welfare under Subadditive Valuations

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'egh, and Vondr'ak (STOC 2023). In addition, we develop a polynomial-time algorithm which, given an arbitrary allocation widetildeA as input, returns an EF1 allocation with NSW at least frac1e2/e≈ frac12.08 times that of widetildeA. 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 frac12-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.