  When selecting committees based on preferences of voters, a variety of
different criteria can be considered. Two natural objectives are maximizing the
utilitarian welfare (the sum of voters' utilities) and coverage (the number of
represented voters) of the selected committee. Previous work has studied the
impact on utilitarian welfare and coverage when requiring the committee to
satisfy minimal requirements such as justified representation or weak
proportionality. In this paper, we consider the impact of imposing much more
demanding proportionality axioms. We identify a class of voting rules that
achieve strong guarantees on utilitarian welfare and coverage when combined
with appropriate completions. This class is defined via a weakening of
priceability and contains prominent rules such as the Method of Equal Shares.
We show that committees selected by these rules (i) can be completed to achieve
optimal coverage and (ii) can be completed to achieve an asymptotically optimal
approximation to the utilitarian welfare if they additionally satisfy EJR+.
Answering an open question of Elkind et al. (2022), we use the Greedy Justified
Candidate Rule to obtain the best possible utilitarian guarantee subject to
proportionality. We also consider completion methods suggested in the
participatory budgeting literature and other objectives besides welfare and
coverage.
