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

Completing Priceable Committees: Utilitarian and Representation Guarantees for Proportional Multiwinner Voting