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

Even though proportional representation is a fundamental goal in multiwinner voting and a plethora of proportionality notions has been introduced, the normative justifications for choosing one notion over another remain poorly understood. We address this by introducing the axiomatic study of proportionality notions in the approval-based multiwinner voting setting. That is, we define axioms (or desirable properties) that ``good'' proportionality notions should possess. Using these axioms, we then provide axiomatic characterizations of two prominent recently introduced notions: PJR+ and EJR+ [Brill and Peters 2023]. Our characterization proceeds in two parts. Firstly, we provide a characterization of refinements of PJR+ and EJR+. That is, we define axioms such that any notion satisfying these axioms must imply PJR+ (or EJR+, respectively). In particular, the fundamental axiom distinguishing PJR+ and EJR+ from their predecessors PJR and EJR is the classical axiom of monotonicity. Secondly, we introduce our framework of witness-based proportionality notions, that is, proportionality notions that certify ``misrepresentation'' via a witness set of misrepresented voters. In this class, we provide characterizations of PJR+ and EJR+ as the strongest (assuming certain axioms). Thus, by putting both directions together we obtain exact characterizations of both notions. Among our results, it may be worth highlighting that any notion satisfying mild conditions (monotonicity, independence of losers, robustness to fully satisfied voters, and lower quota) refines PJR+. In this sense, PJR+ turns out to be the canonical minimal requirement that one may impose on proportionality.