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.

翻译：基于选民偏好选择委员会时，可考虑多种不同标准。两个自然目标包括最大化选定委员会的功利福利（选民效用之和）与覆盖范围（被代表的选民数量）。已有研究探讨了在要求委员会满足公正代表性或弱比例性等最低条件时对功利福利与覆盖范围的影响。本文研究了施加更高要求的比例性公理的影响。我们识别出一类投票规则，当与适当的补全方法结合时，能在功利福利与覆盖范围上实现强保障。此类规则通过价格性的弱化定义，包含等额分配法等著名规则。研究表明：由这些规则选出的委员会（i）可被补全以实现最优覆盖范围，且（ii）若额外满足EJR+公理，可被补全以实现功利福利的渐近最优近似。为回答Elkind等人（2022年）的开放性问题，我们采用贪婪正当候选人规则，在满足比例性的前提下获得最优功利保障。此外，我们还考虑了参与式预算文献中提出的补全方法，以及除福利与覆盖范围之外的其他目标。