  We revisit the setting of fairly allocating indivisible items when agents
have different weights representing their entitlements. First, we propose a
parameterized family of relaxations for weighted envy-freeness and the same for
weighted proportionality; the parameters indicate whether smaller-weight or
larger-weight agents should be given a higher priority. We show that each
notion in these families can always be satisfied, but any two cannot
necessarily be fulfilled simultaneously. We then introduce an intuitive
weighted generalization of maximin share fairness and establish the optimal
approximation of it that can be guaranteed. Furthermore, we characterize the
implication relations between the various weighted fairness notions introduced
in this and prior work, and relate them to the lower and upper quota axioms
from apportionment.
