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Proportional Justified Representation

Luis Sánchez-Fernández, Edith Elkind, Martin Lackner, Norberto Fernández, Jesús A. Fisteus, Pablo Basanta Val, Piotr Skowron

TL;DR

The paper investigates representation in approval-based multi-winner elections, revealing a tension between Perfect Representation (PR) and Extended Justified Representation (EJR). It introduces Proportional Justified Representation (PJR) as a PR-compatible relaxation of EJR and studies its algorithmic and axiomatic properties, including its relationship to Proportional Approval Voting (PAV) and average voter satisfaction. To address limitations of PR, it defines Fractional Perfect Representation (FPR) via a flow formulation, showing PR ⇔ FPR when $k|n$ and that FPR implies PJR, while remaining incompatible with EJR and core. The work also connects these notions to apportionment concepts, priceability, and laminar proportionality, and analyzes the behavior of Monroe's rule under FPR. Overall, it provides a nuanced landscape of representation axioms, highlighting trade-offs between fairness guarantees and computational tractability, with implications for both political and AI-enabled grouping contexts.

Abstract

The goal of multi-winner elections is to choose a fixed-size committee based on voters' preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. In an influential paper, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). We observe that EJR is incompatible with the highly desirable Perfect Representation (PR) criterion, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with PR, as well as with a stronger variant of this axiom, which we term Fractional Perfect Representation (FPR). Moreover, just like EJR, PJR can be used to characterise the classic Proportional Approval Voting (PAV) rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.

Proportional Justified Representation

TL;DR

The paper investigates representation in approval-based multi-winner elections, revealing a tension between Perfect Representation (PR) and Extended Justified Representation (EJR). It introduces Proportional Justified Representation (PJR) as a PR-compatible relaxation of EJR and studies its algorithmic and axiomatic properties, including its relationship to Proportional Approval Voting (PAV) and average voter satisfaction. To address limitations of PR, it defines Fractional Perfect Representation (FPR) via a flow formulation, showing PR ⇔ FPR when and that FPR implies PJR, while remaining incompatible with EJR and core. The work also connects these notions to apportionment concepts, priceability, and laminar proportionality, and analyzes the behavior of Monroe's rule under FPR. Overall, it provides a nuanced landscape of representation axioms, highlighting trade-offs between fairness guarantees and computational tractability, with implications for both political and AI-enabled grouping contexts.

Abstract

The goal of multi-winner elections is to choose a fixed-size committee based on voters' preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. In an influential paper, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). We observe that EJR is incompatible with the highly desirable Perfect Representation (PR) criterion, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with PR, as well as with a stronger variant of this axiom, which we term Fractional Perfect Representation (FPR). Moreover, just like EJR, PJR can be used to characterise the classic Proportional Approval Voting (PAV) rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.

Paper Structure

This paper contains 15 sections, 14 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Fair Representation and AI
  3. Preliminaries
  4. Perfect Representation
  5. Further Examples
  6. Voting on Issues
  7. Participatory Budgeting
  8. Proportional Justified Representation
  9. Average Satisfaction
  10. Fractional Perfect Representation
  11. FPR and the Monroe Rule
  12. FPR and Other Proportionality Axioms
  13. FPR and Individual Representation Axioms
  14. Relationship with apportionment axioms
  15. Discussion

Figures (6)

  • Figure 1: Hasse diagram for the representation axioms considered in this paper and other state-of-the-art papers. The lines indicate that if a committee satisfies the upper axiom, it also satisfies the lower axiom. Axioms in bold are contributions of this paper.
  • Figure 2: Ballot profile in the proof of Theorem \ref{['thm:pr-ejr']}.
  • Figure 3: Ballot profile in Example \ref{['ex:consensus']}.
  • Figure 4: Ballot profile in Example \ref{['ex:three-yes-no']}.
  • Figure 5: Ballot profile in Example \ref{['ex:monroe']}.
  • ...and 1 more figures

Theorems & Definitions (20)

  • proof
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  • ...and 10 more