  We consider the classic problem of fairly allocating indivisible goods among
agents with additive valuation functions and explore the connection between two
prominent fairness notions: maximum Nash welfare (MNW) and envy-freeness up to
any good (EFX). We establish that an MNW allocation is always EFX as long as
there are at most two possible values for the goods, whereas this implication
is no longer true for three or more distinct values. As a notable consequence,
this proves the existence of EFX allocations for these restricted valuation
functions. While the efficient computation of an MNW allocation for two
possible values remains an open problem, we present a novel algorithm for
directly constructing EFX allocations in this setting. Finally, we study the
question of whether an MNW allocation implies any EFX guarantee for general
additive valuation functions under a natural new interpretation of approximate
EFX allocations.
