  We study the efficiency of fair allocations using the well-studied price of
fairness concept, which quantitatively measures the worst-case efficiency loss
when imposing fairness constraints. Previous works provided partial results on
the price of fairness with well-known fairness notions such as envy-freeness up
to one good (EF1) and envy-freeness up to any good (EFX). In this paper, we
give a complete characterization for the price of envy-freeness in various
settings. In particular, we first consider the two-agent case under the
indivisible-goods setting and present tight ratios for the price of EF1 (for
scaled utility) and EFX (for unscaled utility), which resolve questions left
open in the literature. Next, we consider the mixed goods setting which
concerns a mixture of both divisible and indivisible goods. We focus on
envy-freeness for mixed goods (EFM), which generalizes both envy-freeness and
EF1, as well as its strengthening called envy-freeness up to any good for mixed
goods (EFXM), which generalizes envy-freeness and EFX. To this end, we settle
the price of EFM and EFXM by providing a complete picture of tight bounds for
two agents and asymptotically tight bounds for $n$ agents, for both scaled and
unscaled utilities.
