  In fair division problems with indivisible goods it is well known that one
cannot have any guarantees for the classic fairness notions of envy-freeness
and proportionality. As a result, several relaxations have been introduced,
most of which in quite recent works. We focus on four such notions, namely
envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin
share fairness (MMS), and pairwise maximin share fairness (PMMS). Since
obtaining these relaxations also turns out to be problematic in several
scenarios, approximate versions of them have been considered. In this work, we
investigate further the connections between the four notions mentioned above
and their approximate versions. We establish several tight, or almost tight,
results concerning the approximation quality that any of these notions
guarantees for the others, providing an almost complete picture of this
landscape. Some of our findings reveal interesting and surprising consequences
regarding the power of these notions, e.g., PMMS and EFX provide the same
worst-case guarantee for MMS, despite PMMS being a strictly stronger notion
than EFX. We believe such implications provide further insight on the quality
of approximately fair solutions.
