We introduces a general linear framework that unifies the study of multi-winner voting rules and proportionality axioms, demonstrating that many prominent multi-winner voting rules-including Thiele methods, their sequential variants, and approval-based committee scoring rules-are linear. Similarly, key proportionality axioms such as Justified Representation (JR), Extended JR (EJR), and their strengthened variants (PJR+, EJR+), along with core stability, can fit within this linear structure as well. Leveraging PAC learning theory, we establish general and novel upper bounds on the sample complexity of learning linear mappings. Our approach yields near-optimal guarantees for diverse classes of rules, including Thiele methods and ordered weighted average rules, and can be applied to analyze the sample complexity of learning proportionality axioms such as approximate core stability. Furthermore, the linear structure allows us to leverage prior work to extend our analysis beyond worst-case scenarios to study the likelihood of various properties of linear rules and axioms. We introduce a broad class of distributions that extend Impartial Culture for approval preferences, and show that under these distributions, with high probability, any Thiele method is resolute, CORE is non-empty, and any Thiele method satisfies CORE, among other observations on the likelihood of commonly-studied properties in social choice. We believe that this linear theory offers a new perspective and powerful new tools for designing and analyzing multi-winner rules in modern social choice applications.

翻译：本文提出了一种通用的线性框架，该框架统一了对多赢家投票规则与比例性公理的研究，证明了包括蒂勒方法、其序列变体以及基于赞同的委员会评分规则在内的许多重要多赢家投票规则都是线性的。类似地，关键的比例性公理，如合理代表（JR）、扩展合理代表（EJR）及其强化变体（PJR+、EJR+），以及核心稳定性，同样可以纳入这一线性结构中。借助PAC学习理论，我们为学习线性映射的样本复杂度建立了通用且新颖的上界。我们的方法为包括蒂勒方法和有序加权平均规则在内的多种规则类别提供了接近最优的保证，并可应用于分析学习比例性公理（如近似核心稳定性）的样本复杂度。此外，线性结构使我们能够利用先前的研究，将分析从最坏情况扩展到研究线性规则与公理的各种性质出现的可能性。我们引入了一类广泛的分布，扩展了针对赞同偏好的公正文化假设，并证明在这些分布下，以高概率，任何蒂勒方法都是确定的，CORE是非空的，并且任何蒂勒方法都满足CORE，此外还包括对社会选择中常见研究性质出现可能性的其他观察。我们相信，这一线性理论为现代社会选择应用中多赢家规则的设计与分析提供了新的视角和强大的新工具。