  A major problem in fair division is how to allocate a set of indivisible
resources among agents fairly and efficiently. The goal of this work is to
characterize the tradeoffs between two well-studied measures of fairness and
efficiency -- envy freeness up to any item (EFX) for fairness, and Nash welfare
for efficiency -- by saying, for given constants $\alpha$ and $\beta$, whether
there exists an $\alpha$-EFX allocation that guarantees a $\beta$-fraction of
the maximum Nash welfare ($\beta$-MNW). For additive valuations, we show that
for any $\alpha \in [0,1]$, there exists a partial allocation that is
$\alpha$-EFX and $\frac{1}{\alpha+1}$-MNW, and this tradeoff is tight (for any
$\alpha$). We also show that for $\alpha\in[0,\varphi-1 \approx 0.618]$ these
partial allocations can be turned into complete allocations where all items are
assigned. Furthermore, for any $\alpha \in [0, 1/2]$, we show that the tight
tradeoff of $\alpha$-EFX and $\frac{1}{\alpha+1}$-MNW with complete allocations
holds for the more general setting of subadditive valuations. Our results
improve upon the current state of the art, for both additive and subadditive
valuations, and match the best-known approximations of EFX under complete
allocations, regardless of Nash welfare guarantees. Notably, our constructions
for additive valuations also provide EF1 and constant approximations for
maximin share guarantees.
