  To choose a suitable multiwinner voting rule is a hard and ambiguous task.
Depending on the context, it varies widely what constitutes the choice of an
``optimal'' subset of alternatives. In this paper, we provide a quantitative
analysis of multiwinner voting rules using methods from the theory of
approximation algorithms---we estimate how well multiwinner rules approximate
two extreme objectives: a representation criterion defined via the Approval
Chamberlin--Courant rule and a utilitarian criterion defined via Multiwinner
Approval Voting. With both theoretical and experimental methods, we classify
multiwinner rules in terms of their quantitative alignment with these two
opposing objectives. Our results provide fundamental information about the
nature of multiwinner rules and, in particular, about the necessary tradeoffs
when choosing such a rule.
