  We study multiwinner elections with approval-based preferences. An instance
of a multiwinner election consists of a set of alternatives, a population of
voters---each voter approves a subset of alternatives, and the desired
committee size $k$; the goal is to select a committee (a~subset) of $k$
alternatives according to the preferences of the voters. We investigate a
number of election rules and ask whether the committees that they return
represent the voters proportionally. In contrast to the classic literature, we
employ quantitative techniques that allow to measure the extent to which the
considered rules are proportional. This allows us to arrange the rules in a
clear hierarchy. For example, we find that Proportional Approval Voting (PAV)
has better proportionality guarantees than its sequential counterpart, and that
Phragm\'{e}n's Sequential Rule is worse than Sequential PAV. Yet, the loss of
proportionality for the two sequential rules is moderate and in some contexts
can be outweighed by their other appealing properties. Finally, we measure the
tradeoff between proportionality and utilitarian efficiency for a broad
subclass of committee election rules.
