The Price of Justified Representation

TL;DR

This paper investigates the trade-offs and computational challenges of enforcing justified representation in multiwinner approval voting, analyzing its impact on social welfare and coverage through theoretical bounds, algorithms, and empirical data.

Contribution

It provides worst-case bounds, complexity results, algorithms, and empirical analysis of the effects of justified representation in approval voting.

Findings

Imposing JR can significantly reduce social welfare and coverage.

Finding committees that satisfy JR is computationally hard, but approximation algorithms exist.

Empirical results show the practical impact of JR constraints on committee quality.

Abstract

In multiwinner approval voting, the goal is to select $k$-member committees based on voters' approval ballots. A well-studied concept of proportionality in this context is the justified representation (JR) axiom, which demands that no large cohesive group of voters remains unrepresented. However, the JR axiom may conflict with other desiderata, such as coverage (maximizing the number of voters who approve at least one committee member) or social welfare (maximizing the number of approvals obtained by committee members). In this work, we investigate the impact of imposing the JR axiom (as well as the more demanding EJR axiom) on social welfare and coverage. Our approach is threefold: we derive worst-case bounds on the loss of welfare/coverage that is caused by imposing JR, study the computational complexity of finding 'good' committees that provide JR (obtaining a hardness result, an approximation algorithm, and an exact algorithm for one-dimensional preferences), and examine this setting empirically on several synthetic datasets.