A major problem in fair division is how to allocate a set of indivisible resources among agents fairly and efficiently. We give optimal tradeoffs between fairness and efficiency, with respect to well-studied measures of fairness and efficiency -- envy freeness up to any item (EFX) for fairness, and Nash welfare for efficiency. Our results improve upon the current state of the art, for both additive and subadditive valuations. For additive valuations, we show the existence of allocations that are simultaneously $\alpha$-EFX and guarantee a $\frac{1}{\alpha+1}$-fraction of the maximum Nash welfare, for any $\alpha\in[0,1]$. For $\alpha\in[0,\varphi-1 \approx 0.618]$ these are complete allocations (all items are assigned), whereas for larger $\alpha$ these are partial allocations (some items may be unassigned). We partially extend this to subadditive valuations where we show the existence of complete allocations that give $\alpha$-EFX and a $\frac{1}{\alpha+1}$-fraction of the maximum Nash welfare (as above), for any $\alpha\in[0,1/2]$. We also give impossibility results that show that our tradeoffs are tight, even with respect to partial allocations.

翻译：公平分配中的一个主要问题是如何在公平和高效之间分配一组不可分割资源。我们针对公平和效率的经典度量——公平性方面达到任意物品无嫉妒（EFX），效率方面达到纳什福利——给出了公平与效率之间的最优权衡。我们的结果改进了现有技术水平，适用于加性估值和次加性估值。对于加性估值，我们证明了存在同时为α-EFX并保证最大纳什福利的1/(α+1)倍份额的分配方案，其中α∈[0,1]。当α∈[0, φ-1 ≈ 0.618]时，这些分配是完全分配（所有物品均被分配）；而对于更大的α，则为部分分配（部分物品可能未被分配）。我们将此部分扩展到次加性估值，证明了对于任意α∈[0,1/2]，存在完全分配方案，满足α-EFX且达到最大纳什福利的1/(α+1)倍（同上）。我们还给出了不可能性结果，表明我们的权衡是最优的，即使考虑部分分配也是如此。