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

TL;DR

This paper explores how to select committees that satisfy strong proportionality axioms while maximizing utilitarian welfare and coverage, introducing a class of rules that achieve these guarantees through committee completion.

Contribution

It identifies a new class of voting rules based on weakened priceability that guarantees strong welfare and coverage when completed, addressing open questions in proportional multiwinner voting.

Findings

Committee rules can be completed to achieve optimal coverage.

Rules can be completed to approximate utilitarian welfare asymptotically.

Greedy Justified Candidate Rule provides the best utilitarian guarantee under proportionality.

Abstract

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.