  The goal of multi-winner elections is to choose a fixed-size committee based
on voters' preferences. An important concern in this setting is representation:
large groups of voters with cohesive preferences should be adequately
represented by the election winners. Recently, Aziz et al. (2015a;2017)
proposed two axioms that aim to capture this idea: justified representation
(JR) and its strengthening extended justified representation (EJR). In this
paper, we extend the work of Aziz et al. in several directions. First, we
answer an open question of Aziz et al., by showing that Reweighted Approval
Voting satisfies JR for $k=3, 4, 5$, but fails it for $k\ge 6$. Second, we
observe that EJR is incompatible with the Perfect Representation criterion,
which is important for many applications of multi-winner voting, and propose a
relaxation of EJR, which we call Proportional Justified Representation (PJR).
PJR is more demanding than JR, but, unlike EJR, it is compatible with perfect
representation, and a committee that provides PJR can be computed in polynomial
time if the committee size divides the number of voters. Moreover, just like
EJR, PJR can be used to characterize the classic PAV rule in the class of
weighted PAV rules. On the other hand, we show that EJR provides stronger
guarantees with respect to average voter satisfaction than PJR does.
