[Paper Review] Proportional Justified Representation

Luis Sánchez Fernández, Edith Elkind+5 more|arXiv (Cornell University)|Jan 1, 2016

Game Theory and Voting SystemsEconomics, Econometrics and Finance24 citations

TL;DR

This paper introduces Proportional Justified Representation (PJR), a relaxed yet compatible alternative to Extended Justified Representation (EJR), resolving incompatibility with Perfect Representation. It proves that Reweighted Approval Voting satisfies JR for committee sizes k = 3, 4, 5 but fails for k ≥ 6, and shows that PJR enables polynomial-time computation of proportional committees when the committee size divides the number of voters.

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. Recently, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). In this paper, we extend the work of Aziz et al. in several directions. First, we answer an open question of Aziz et al., by showing that Reweighted Approval Voting satisfies JR for k = 3; 4; 5, but fails it for k >= 6. Second, we observe that EJR is incompatible with the Perfect Representation criterion, which is important for many applications of multi-winner voting, 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 perfect representation, and a committee that provides PJR can be computed in polynomial time if the committee size divides the number of voters. Moreover, just like EJR, PJR can be used to characterize the classic 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.

Motivation & Objective

• To resolve the incompatibility between Extended Justified Representation (EJR) and the Perfect Representation criterion in multi-winner elections.
• To propose a new axiom, Proportional Justified Representation (PJR), that strengthens JR while remaining compatible with perfect representation.
• To establish that PJR can be computed in polynomial time when the committee size divides the number of voters.
• To characterize the classic Proportional Approval Voting (PAV) rule within the class of weighted PAV rules using PJR.
• To compare the voter satisfaction guarantees of EJR and PJR, showing EJR provides stronger average satisfaction guarantees.

Proposed method

• Proposes Proportional Justified Representation (PJR) as a relaxation of EJR that preserves compatibility with the Perfect Representation criterion.
• Analyzes the behavior of Reweighted Approval Voting (RAV) under different committee sizes, proving it satisfies JR for k = 3, 4, 5 but fails for k ≥ 6.
• Demonstrates that a committee satisfying PJR can be computed in polynomial time when the committee size divides the number of voters.
• Uses axiomatic analysis to show that PJR characterizes the classic PAV rule among weighted PAV rules.
• Compares EJR and PJR via theoretical guarantees on average voter satisfaction, showing EJR offers stronger performance in this regard.

Experimental results

Research questions

• RQ1 Is Extended Justified Representation (EJR) compatible with the Perfect Representation criterion in multi-winner elections?
• RQ2 Can a stronger yet compatible alternative to JR be formulated that preserves desirable computational and representational properties?
• RQ3 For which committee sizes does Reweighted Approval Voting satisfy the Justified Representation (JR) axiom?
• RQ4 Does Proportional Justified Representation (PJR) enable polynomial-time computation of proportional outcomes when the committee size divides the number of voters?
• RQ5 How do EJR and PJR compare in terms of average voter satisfaction guarantees?

Key findings

• EJR is incompatible with the Perfect Representation criterion, which limits its applicability in settings requiring perfect representation.
• PJR is proposed as a relaxation of EJR that is compatible with Perfect Representation while still being more demanding than JR.
• Reweighted Approval Voting satisfies JR for committee sizes k = 3, 4, 5 but fails to satisfy JR for k ≥ 6.
• A committee satisfying PJR can be computed in polynomial time when the committee size divides the number of voters.
• PJR characterizes the classic Proportional Approval Voting (PAV) rule within the class of weighted PAV rules.
• EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.