When selecting a subset of candidates (a so-called committee) based on the preferences of voters, proportional representation is often a major desideratum. When going beyond simplistic models such as party-list or district-based elections, it is surprisingly challenging to capture proportionality formally. As a consequence, the literature has produced numerous competing criteria of when a selected committee qualifies as proportional. Two of the most prominent notions are Dummett's proportionality for solid coalitions (PSC) and Aziz et al.'s extended justified representation (EJR). Both guarantee proportional representation to groups of voters who have very similar preferences; such groups are referred to as solid coalitions by Dummett and as cohesive groups by Aziz et al. However, these notions lose their bite when groups are only almost solid or almost cohesive. In this paper, we propose proportionality axioms that are more robust: they guarantee representation also to groups that do not qualify as solid or cohesive. Further, our novel axioms can be easily verified: Given a committee, we can check in polynomial time whether it satisfies the axiom or not. This is in contrast to many established notions like EJR, for which the corresponding verification problem is known to be intractable. In the setting with approval preferences, we propose a robust and verifiable variant of EJR and a simply greedy procedure to compute committees satisfying it. In the setting with ranked preferences, we propose a robust variant PSC, which can be efficiently verified even for general weak preferences. In the special case of strict preferences, our notion is the first known satisfiable proportionality axiom that is violated by the Single Transferable Vote (STV). We also discuss implications of our results for participatory budgeting, querying procedures, and to the notion of proportionality degree.

翻译：基于选民偏好选择候选子集（即委员会）时，比例代表制通常是重要目标。当超越政党名单制或选区选举等简化模型时，正式刻画比例性变得异常困难。因此，学界提出了大量相互竞争的准则来判定所选委员会是否具备比例性。其中最著名的两个概念是达梅特的固态联盟比例性(PSC)和阿齐兹等人的扩展公正代表性(EJR)。二者均保证偏好高度相似的选民群体获得比例代表权——达梅特称此类群体为固态联盟，阿齐兹等人则称其为凝聚性群体。然而，当群体仅近乎固态或近乎凝聚时，这些概念便失去效力。本文提出更具鲁棒性的比例性公理：即使群体不具备固态或凝聚性特征，仍能保障其代表性。此外，新公理易于验证：给定委员会，可在多项式时间内检验其是否满足该公理。这与EJR等许多既有概念形成鲜明对比，后者对应的验证问题已被证明难以求解。在批准投票偏好设定中，我们提出EJR的鲁棒可验证变体及简单的贪心算法来构造满足该公理的委员会。在排序偏好设定中，我们提出PSC的鲁棒变体，即便面对一般弱偏好也能高效验证。在严格偏好的特殊情形下，该概念是首个已知的可满足比例性公理，却能被单一可转移投票(STV)违反。最后讨论本文结果对参与式预算、查询程序及比例性度量的启示。