An Axiomatic Analysis of Proportionality Notions in Approval-Based Multiwinner Voting

Researchers Chris Dong and Jannik Peters have published an axiomatic analysis of proportionality notions in approval-based multiwinner voting, submitted to arXiv in May 2026. Their work addresses the normative gap in choosing between competing proportionality standards like PJR, EJR, PJR+, and EJR+. They define a set of desirable axioms—monotonicity, independence of losers, robustness to fully satisfied voters, and lower quota—that any 'good' proportionality notion should satisfy. Their main result shows that any notion satisfying these mild conditions necessarily refines PJR+, establishing PJR+ as the canonical minimal requirement for proportionality in approval-based multiwinner voting.

The authors provide a two-part characterization. First, they show that refinements of PJR+ and EJR+ are uniquely identified by the classical monotonicity axiom, which distinguishes these newer notions from their predecessors PJR and EJR. Second, they introduce a framework of witness-based proportionality notions, where misrepresentation is certified by a witness set of voters. Within this framework, they characterize PJR+ and EJR+ as the strongest notions under the given axioms. The paper also notes that independence of losers and robustness to fully satisfied voters play key roles in the characterization. This work gives committee designers a clear, axiom-driven rationale for adopting PJR+ as the minimal standard, and it opens the door for further refinement based on additional normative preferences.