Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Refined Characterizations of Approval-based Committee Scoring Rules

Chris Dong, Patrick Lederer

TL;DR

This work delivers full axiomatic characterizations of two central classes of approval-based committee (ABC) scoring rules—Thiele rules and ballot size weighted approval voting (BSWAV)—within the standard ABC election model, thereby extending Lackner and Skowron’s ranking-based results to the standard setting. By employing consistency for variable electorates and introducing the choice set convexity axiom, the authors prove that Thiele rules are exactly those anonymous, neutral, consistent, continuous, and independent of losers, while BSWAV rules are exactly those anonymous, neutral, consistent, continuous, weakly efficient, and choice set convex. The paper further derives full characterizations for AV, PAV, and SAV, showing AV as the intersection of the two classes under certain conditions, PAV via party-proportionality within Thiele, and SAV via party-proportionality plus aversion to unanimous committees within BSWAV. These results provide modular, widely applicable tools for analyzing and comparing ABC scoring rules, with implications for fairness, proportionality, and computational aspects in multiwinner elections.

Abstract

In approval-based committee (ABC) elections, the goal is to select a fixed-size subset of the candidates, a so-called committee, based on the voters' approval ballots over the candidates. One of the most popular classes of ABC voting rules are ABC scoring rules, which have recently been characterized by Lackner and Skowron (2021). However, this characterization relies on a model where the output is a ranking of committees instead of a set of winning committees and no full characterization of ABC scoring rules exists in the latter standard setting. We address this issue by characterizing two important subclasses of ABC scoring rules in the standard ABC election model, thereby both extending the result of Lackner and Skowron (2021) to the standard setting and refining it to subclasses. In more detail, by relying on a consistency axiom for variable electorates, we characterize (i) the prominent class of Thiele rules and (ii) a new class of ABC voting rules called ballot size weighted approval voting. Based on these theorems, we also infer characterizations of three well-known ABC voting rules, namely multi-winner approval voting, proportional approval voting, and satisfaction approval voting.

Refined Characterizations of Approval-based Committee Scoring Rules

TL;DR

This work delivers full axiomatic characterizations of two central classes of approval-based committee (ABC) scoring rules—Thiele rules and ballot size weighted approval voting (BSWAV)—within the standard ABC election model, thereby extending Lackner and Skowron’s ranking-based results to the standard setting. By employing consistency for variable electorates and introducing the choice set convexity axiom, the authors prove that Thiele rules are exactly those anonymous, neutral, consistent, continuous, and independent of losers, while BSWAV rules are exactly those anonymous, neutral, consistent, continuous, weakly efficient, and choice set convex. The paper further derives full characterizations for AV, PAV, and SAV, showing AV as the intersection of the two classes under certain conditions, PAV via party-proportionality within Thiele, and SAV via party-proportionality plus aversion to unanimous committees within BSWAV. These results provide modular, widely applicable tools for analyzing and comparing ABC scoring rules, with implications for fairness, proportionality, and computational aspects in multiwinner elections.

Abstract

In approval-based committee (ABC) elections, the goal is to select a fixed-size subset of the candidates, a so-called committee, based on the voters' approval ballots over the candidates. One of the most popular classes of ABC voting rules are ABC scoring rules, which have recently been characterized by Lackner and Skowron (2021). However, this characterization relies on a model where the output is a ranking of committees instead of a set of winning committees and no full characterization of ABC scoring rules exists in the latter standard setting. We address this issue by characterizing two important subclasses of ABC scoring rules in the standard ABC election model, thereby both extending the result of Lackner and Skowron (2021) to the standard setting and refining it to subclasses. In more detail, by relying on a consistency axiom for variable electorates, we characterize (i) the prominent class of Thiele rules and (ii) a new class of ABC voting rules called ballot size weighted approval voting. Based on these theorems, we also infer characterizations of three well-known ABC voting rules, namely multi-winner approval voting, proportional approval voting, and satisfaction approval voting.
Paper Structure (31 sections, 27 theorems, 6 equations, 1 figure)

This paper contains 31 sections, 27 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Related work.
  4. Preliminaries
  5. ABC Voting Rules
  6. ABC scoring rules.
  7. Thiele rules.
  8. BSWAV rules.
  9. Basic Axioms
  10. Anonymity.
  11. Neutrality.
  12. Weak Efficiency.
  13. Continuity.
  14. Consistency.
  15. Independence of Losers.
  16. ...and 16 more sections

Key Result

Theorem 1

An ABC voting rule is a Thiele rule if and only if it satisfies anonymity, neutrality, consistency, continuity, and independence of losers.

Figures (1)

  • Figure 1: Overview of our results. An arrow from $X$ to $Y$ means that $Y$ is a subset or an element of $X$. The axioms written on an arrow from $X$ to $Y$ characterize the rule $Y$ within the class $X$. The axioms written below Thiele rules and BSWAV rules characterize these classes of ABC voting rules.

Theorems & Definitions (54)

  • Theorem 1
  • proof : Proof Sketch
  • Theorem 2
  • proof : Proof Sketch
  • Proposition 1
  • proof
  • Proposition 2
  • proof : Proof Sketch
  • Proposition 3
  • proof : Proof Sketch
  • ...and 44 more