Explanation Systems for Approval-Based Multiwinner Voting

TL;DR

This paper introduces a framework for explaining how approval-based multiwinner voting committees represent voters, using price systems to quantify influence, backed by axiomatic analysis and polynomial-time algorithms.

Contribution

It proposes price systems as a novel explanation method for committee influence, along with axioms and algorithms to compute and analyze these explanations.

Findings

Price systems effectively quantify voter influence in committees.

The proposed algorithm satisfies multiple axioms of structural coherence and proportionality.

Experiments show explanations align with established proportionality notions and detect unequal influence.

Abstract

In approval-based multiwinner voting, voters express approval preferences over a set of candidates, and the goal is to return a winning committee. This model captures a broad range of subset selection problems under preferences. Prior work has focused on the study of binary proportionality axioms that certify whether a given committee is proportionally representative or not. We take a more fine-grained perspective and initiate the study of explanation systems that quantify how a committee represents the electorate, i.e., how much influence each voter exerts, how this influence is allocated across selected candidates, how each candidate is backed by the voters, and why certain candidates were not chosen. Building on the notion of priceability, we propose price systems as a framework for such explanations. A price system assigns each voter an individual budget, which they can spend on selected candidates they approve, and each candidate needs to be purchased at a unit price. Since many price systems can exist for a given outcome, selecting among them requires care. We initiate an axiomatic study of price systems and propose several axioms capturing structural coherence, faithful attribution of influence, and alignment with proportionality. On the algorithmic side, we introduce a polynomial-time computable rule in which voters continuously gain and exercise influence and show that it satisfies all jointly satisfiable axioms. Experiments on synthetic and real-world instances indicate that our explanations correlate with established proportionality notions and can recover unequal influence when it is present.