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Allocation Proportionality of OWA--Based Committee Scoring Rules

Daria Boratyn, Dariusz Stolicki

Abstract

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.

Allocation Proportionality of OWA--Based Committee Scoring Rules

Abstract

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, -Borda, Chamberlin-Courant, Harmonic Borda, Proportional -Approval Voting, and Bloc Voting.
Paper Structure (18 sections, 2 theorems, 19 equations, 4 figures)

This paper contains 18 sections, 2 theorems, 19 equations, 4 figures.

Table of Contents

  1. Preliminaries
  2. Motivation
  3. Contribution
  4. Notation
  5. Proportionality Axioms and Quantitative Measures
  6. Proportionality Axioms
  7. Quantitative Measures
  8. Theoretical Framework: Party Elections
  9. Committee Scoring Rules
  10. Allocation Proportionality
  11. Aggregate party scores $\psi$ (Def. \ref{['def:aps']}).
  12. Seat shares under common OWA rules.
  13. Deviation from allocation proportionality.
  14. Experiments
  15. Statistical Cultures
  16. ...and 3 more sections

Key Result

Lemma 1

Fix an ordinal party election $(C,P,a,L_C,w)$ and a scoring vector $\mathbf{s}$. Fix any OWA--based rule $\mathcal{R}$ induced by $\mathbf{s}$. Then:

Figures (4)

  • Figure 1: Allocation proportionality degree ($\alpha$-divergence).
  • Figure 2: Allocation proportionality degree ($L_2$ metric).
  • Figure 3: Defractionalization (change in the effective number of parties).
  • Figure 4: Bias for the largest party.

Theorems & Definitions (28)

  • Definition 1: Party Election
  • Definition 2: Multi-District Party Election
  • Definition 3: Position Vector
  • Definition 4: Partial Order on Position Vectors
  • Definition 5: Committee Scoring Function ElkindEtAl17a
  • Definition 6: Committee Scoring Rules ElkindEtAl17a
  • Definition 7: OWA--Based Scoring Rule
  • Definition 8: SNTV
  • Definition 9: $k$--Borda
  • Definition 10: Bloc Voting
  • ...and 18 more