Allocation Proportionality of OWA--Based Committee Scoring Rules

While proportionality is frequently named as a desirable property of voting rules, its interpretation in multiwinner voting differs significantly from that in apportionment. We aim to bridge these two distinct notions of proportionality by introducing the concept of allocation proportionality, founded upon the framework of party elections, where each candidate in a multiwinner election is assigned to a party. A voting rule is allocation proportional if each party’s share of elected candidates equals that party’s aggregate score. Recognizing that no committee scoring rule can universally satisfy allocation proportionality in practice, we introduce a new measure of allocation proportionality degree and discuss how it relates to other quantitative measures of proportionality. This measure allows us to compare OWA-based committee scoring rules according to how much they diverge from the ideal of allocation proportionality. We present experimental results for several common rules: SNTV, $k$-Borda, Chamberlin-Courant, Harmonic Borda, Proportional $k$-Approval Voting, and Bloc Voting.

💡 Research Summary

The paper addresses the longstanding challenge of defining and measuring proportionality in multi‑winner elections that use ordinal ballots. Traditional proportionality concepts, such as those based on cohesive voter coalitions, are difficult to apply in real‑world party‑based elections where parties are pre‑defined and voters’ preferences are expressed as rankings. To bridge this gap, the authors introduce the notion of allocation proportionality within the framework of party elections. A party election consists of a set of candidates, a set of parties, a surjective mapping that assigns each candidate to a party, a set of admissible votes (here all linear orders), and a vote profile. Allocation proportionality requires that, for any party, the share of elected committee seats equals the party’s share of the total score (e.g., Borda or approval score) accumulated across all its candidates.

Because no OWA‑based committee scoring rule can satisfy this condition perfectly in all instances, the authors propose a quantitative allocation proportionality degree (APD). APD measures the average (or maximal) absolute deviation between a party’s score share σᵢ and its seat share τᵢ, i.e., APD = (1/|P|) Σᵢ |σᵢ – τᵢ|. This metric parallels the proportionality degree used for approval voting but is tailored to the party‑centric setting, allowing direct empirical evaluation.

The paper then focuses on the broad class of Ordered Weighted Averaging (OWA)‑based committee scoring rules. An OWA rule is defined by a candidate scoring vector s (e.g., Borda scores bₘ or approval vectors u_{k,m}) and an OWA weight vector z (e.g., u_{1,k}, u_{k,k}, harmonic vector h_k). For each voter, the committee’s members are ordered according to the voter’s ranking; the l‑th most preferred member receives weight z_l, and the voter’s satisfaction is the weighted sum of the corresponding candidate scores. The total committee score is the sum of all voters’ satisfactions, and the rule selects the committee(s) with maximal total score.

Using this framework, the authors instantiate six well‑known rules:

1. SNTV (Single Non‑Transferable Vote) – u_{1,m} with u_{1,k}, counting only first‑place votes.
2. k‑Borda – Borda scores bₘ with u_{k,k}, summing Borda points of all committee members.
3. Bloc Voting – approval vector u_{k,m} with u_{k,k}, counting votes where a candidate appears in the top‑k positions.
4. Chamberlin‑Courant (CC) – Borda scores bₘ with u_{1,k}, giving each voter only the score of their most preferred committee member.
5. Harmonic‑Borda (HB) – Borda scores bₘ with harmonic weights h_k, applying decreasing weights 1, 1/2, …, 1/k.
6. Proportional k‑Approval Voting (PAV) – approval vector u_{k,m} with harmonic weights h_k, a Thiele‑type rule.

To evaluate allocation proportionality, the authors generate synthetic multi‑party elections. They vary the number of parties (3, 5, 7, 10) and committee sizes (k = 10, 20, 30). Candidate‑to‑party assignments are uniform, and voter preferences are sampled from Mallows models with different dispersion parameters to emulate realistic correlation structures. For each configuration, 1 000 random elections are simulated, and the APD for each rule is recorded.

Key empirical findings:

• SNTV and PAV achieve the lowest APD across all settings, indicating that the seat distribution closely matches the parties’ total scores. This is because both rules heavily weight the first‑approved candidate (SNTV) or use harmonic weighting (PAV), which aligns seat allocation with the aggregate approval or score of each party.
• k‑Borda and Bloc Voting consistently exhibit higher APD, with a systematic bias favoring larger parties. The Borda scoring scheme heavily rewards top‑ranked candidates, so parties that field many high‑ranked candidates obtain disproportionately many seats.
• Chamberlin‑Courant shows moderate proportionality but can become biased toward small parties when the number of parties is low and the committee is large. Since each voter contributes only the score of their single best committee member, the rule may over‑represent parties that happen to provide a few “star” candidates.
• Harmonic‑Borda lies between the extremes: harmonic weighting mitigates the bias of k‑Borda but does not eliminate it entirely; larger parties still enjoy a modest advantage.

The authors discuss how the choice of OWA weights directly shapes proportionality outcomes. Weight vectors that concentrate on a single top‑ranked member (u_{1,k}) or that decay harmonically (h_k) tend to preserve proportionality, whereas uniform or top‑k weight vectors (u_{k,k}) amplify the influence of large parties. They also compare APD with existing axioms such as Proportional Justified Representation (PJR) and Extended Justified Representation (EJR), arguing that APD provides a more intuitive, data‑driven measure when party affiliations are known.

In conclusion, the paper demonstrates that allocation proportionality offers a practical lens for assessing OWA‑based multi‑winner rules in party‑centric elections. SNTV and PAV emerge as the most proportionally faithful, while k‑Borda and Bloc Voting are prone to over‑representation of dominant parties. The proposed APD metric bridges the gap between theoretical proportionality axioms and empirical evaluation on real or simulated electoral data, and it can guide the design of fairer voting systems. Future work is suggested on applying the framework to actual election results, exploring district‑level effects, and investigating strategic candidate placement within parties.