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Understanding the Impact of Proportionality in Approval-Based Multiwinner Elections

Niclas Boehmer, Lara Glessen, Jannik Peters

TL;DR

This paper addresses the problem of understanding how proportionality axioms shape outcomes in approval-based multiwinner elections. It combines algorithmic analyses—establishing NP-hardness and fixed-parameter tractability results, and presenting ILP and sampling approaches—with a large-scale experimental study on real-world and synthetic data. It introduces candidate-importance measures (JR-prevalence, EJR+-prevalence, and their power-indices) and analyzes their relationship to traditional approval strength, showing that approval scores alone can be misleading for proportionality. The findings reveal substantial variability in axiom restrictiveness across instances, including real-world cases with strong constraints, and demonstrate that proportional committees are diverse and that certain voting rules (e.g., MES) tend to favor broadly popular candidates over those crucial for proportionality, highlighting practical implications for designing proportional multiwinner rules.

Abstract

Despite extensive theoretical research on proportionality in approval-based multiwinner voting, its impact on which committees and candidates can be selected in practice remains poorly understood. We address this gap by (i) analyzing the computational complexity of several natural problems related to the behavior of proportionality axioms, and (ii) conducting an extensive experimental study on both real-world and synthetic elections. Our findings reveal substantial variation in the restrictiveness of proportionality across instances, including previously unobserved high levels of restrictiveness in some real-world cases. We also introduce and evaluate new measures for quantifying a candidate's importance for achieving proportional outcomes, which differ clearly from assessing candidate strength by approval score.

Understanding the Impact of Proportionality in Approval-Based Multiwinner Elections

TL;DR

This paper addresses the problem of understanding how proportionality axioms shape outcomes in approval-based multiwinner elections. It combines algorithmic analyses—establishing NP-hardness and fixed-parameter tractability results, and presenting ILP and sampling approaches—with a large-scale experimental study on real-world and synthetic data. It introduces candidate-importance measures (JR-prevalence, EJR+-prevalence, and their power-indices) and analyzes their relationship to traditional approval strength, showing that approval scores alone can be misleading for proportionality. The findings reveal substantial variability in axiom restrictiveness across instances, including real-world cases with strong constraints, and demonstrate that proportional committees are diverse and that certain voting rules (e.g., MES) tend to favor broadly popular candidates over those crucial for proportionality, highlighting practical implications for designing proportional multiwinner rules.

Abstract

Despite extensive theoretical research on proportionality in approval-based multiwinner voting, its impact on which committees and candidates can be selected in practice remains poorly understood. We address this gap by (i) analyzing the computational complexity of several natural problems related to the behavior of proportionality axioms, and (ii) conducting an extensive experimental study on both real-world and synthetic elections. Our findings reveal substantial variation in the restrictiveness of proportionality across instances, including previously unobserved high levels of restrictiveness in some real-world cases. We also introduce and evaluate new measures for quantifying a candidate's importance for achieving proportional outcomes, which differ clearly from assessing candidate strength by approval score.

Paper Structure

This paper contains 44 sections, 18 theorems, 16 equations, 17 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Preliminaries
  5. Model and Notation.
  6. Proportionality Axioms.
  7. Parameterized Algorithms.
  8. Measures of Candidate Importance for Proportionality.
  9. Algorithmic Analysis
  10. Differentiating Proportionality Axioms
  11. Distances Between Proportional Committees
  12. Candidate Containment
  13. Counting Committees
  14. Experiments
  15. Dataset.
  16. ...and 29 more sections

Key Result

Theorem 3.1

JR-not-EJR+ is NP-complete, and W[1]-hard when parameterized by the committee size $k$.

Figures (17)

  • Figure 1: Percentage of committees satisfying JR and EJR+ depending on the committee size $k$, in the Warsaw Praga-Północ PB election 2022.
  • Figure 2: Example instance to illustrate JR and EJR+.
  • Figure 3: JR-fraction (red) and EJR+-fraction (blue) for different committee sizes $k\in [m]$ across selected pabulib instances.
  • Figure 4: Number of pabulib instances (in total 360) with given JR-fractions (red) and EJR+-fractions (blue).
  • Figure 5: Each point represents one instance and one axiom (JR in red, EJR+ in blue). We plot the correlation between the axiom fraction and the normalized average distance among committees fulfilling the axiom.
  • ...and 12 more figures

Theorems & Definitions (29)

  • Definition 2.1: EJR+
  • Example 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Proposition 3.7
  • ...and 19 more