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Proportionality and Strategyproofness in Multiwinner Elections

Dominik Peters

TL;DR

It is proved that no multiwinner voting rule can simultaneously satisfy a weak form of proportionality (a weakening of justified representation) and aweak form of strategyproofness in the domain of committee elections.

Abstract

Multiwinner voting rules can be used to select a fixed-size committee from a larger set of candidates. We consider approval-based committee rules, which allow voters to approve or disapprove candidates. In this setting, several voting rules such as Proportional Approval Voting (PAV) and Phragmén's rules have been shown to produce committees that are proportional, in the sense that they proportionally represent voters' preferences; all of these rules are strategically manipulable by voters. On the other hand, a generalisation of Approval Voting gives a non-proportional but strategyproof voting rule. We show that there is a fundamental tradeoff between these two properties: we prove that no multiwinner voting rule can simultaneously satisfy a weak form of proportionality (a weakening of justified representation) and a weak form of strategyproofness. Our impossibility is obtained using a formulation of the problem in propositional logic and applying SAT solvers; a human-readable version of the computer-generated proof is obtained by extracting a minimal unsatisfiable set (MUS). We also discuss several related axiomatic questions in the domain of committee elections.

Proportionality and Strategyproofness in Multiwinner Elections

TL;DR

It is proved that no multiwinner voting rule can simultaneously satisfy a weak form of proportionality (a weakening of justified representation) and aweak form of strategyproofness in the domain of committee elections.

Abstract

Multiwinner voting rules can be used to select a fixed-size committee from a larger set of candidates. We consider approval-based committee rules, which allow voters to approve or disapprove candidates. In this setting, several voting rules such as Proportional Approval Voting (PAV) and Phragmén's rules have been shown to produce committees that are proportional, in the sense that they proportionally represent voters' preferences; all of these rules are strategically manipulable by voters. On the other hand, a generalisation of Approval Voting gives a non-proportional but strategyproof voting rule. We show that there is a fundamental tradeoff between these two properties: we prove that no multiwinner voting rule can simultaneously satisfy a weak form of proportionality (a weakening of justified representation) and a weak form of strategyproofness. Our impossibility is obtained using a formulation of the problem in propositional logic and applying SAT solvers; a human-readable version of the computer-generated proof is obtained by extracting a minimal unsatisfiable set (MUS). We also discuss several related axiomatic questions in the domain of committee elections.

Paper Structure

This paper contains 16 sections, 10 theorems, 2 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Our Axioms
  4. Strategyproofness
  5. Proportionality
  6. Efficiency
  7. The Computer-Aided Approach
  8. The Impossibility Theorem
  9. Base case
  10. Induction steps
  11. Extension to other electorate sizes
  12. Remark.
  13. Small committees
  14. Related Work
  15. Conclusions and Future Work
  16. ...and 1 more sections

Key Result

Theorem 5.1

Suppose $k \geqslant 3$, the number $n$ of voters is divisible by $k$, and $m \geqslant k+1$. Then there exists no approval-based committee rule which satisfies weak efficiency, proportionality and strategyproofness.

Figures (1)

  • Figure 1: Proportionality axioms and logical implications.

Theorems & Definitions (10)

  • Theorem 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Lemma 5.4
  • Lemma 5.5
  • Lemma 5.6
  • Proposition 5.7
  • Theorem 5.8
  • Proposition 5.9
  • Theorem 6.1: Duddy duddy