Proportional Justified Representation

TL;DR

This paper introduces Proportional Justified Representation (PJR), a new fairness axiom for multi-winner elections that balances representation and compatibility with perfect representation, extending prior axioms and characterizing voting rules.

Contribution

It proposes PJR as a relaxation of EJR that is compatible with perfect representation, and characterizes PAV rules within weighted PAV rules.

Findings

PJR is more demanding than JR but compatible with PR and FPR.

EJR guarantees stronger voter satisfaction than PJR.

PJR characterizes the class of weighted PAV voting rules.

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.