A Linear Theory of Multi-Winner Voting
Lirong Xia
TL;DR
The paper introduces a linear framework that unifies multi-winner voting rules and proportionality axioms by representing profiles with Hist$(P)$ and decisions via a finite set of separating hyperplanes. It shows that many existing rules (including Thiele methods and ABCS rules) and axioms (JR, JR+, EJR, CORE, etc.) are linear mappings, enabling PAC-learning analyses and polyhedral likelihood studies under independent approval models. The authors develop compact parameterized subclasses (parameterized maximizers and parameterized hyperplanes) to bound the Natarajan dimension and derive practical sample-complexity guarantees, including polynomial bounds for approximate CORE. They further provide likelihood results showing that CORE is non-empty with high probability under IC-type distributions, and that Thiele methods satisfy CORE with high probability, alongside GST-based dichotomies that govern axiom-rule interactions. Together, these results offer a principled, data-driven lens for designing and analyzing multi-winner rules with provable probabilistic guarantees in modern social choice applications.
Abstract
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.