# Consistent Approval-Based Multi-Winner Rules

## TL;DR

This paper provides an axiomatic analysis of approval-based multi-winner voting rules, characterizing key rules like Proportional Approval Voting and Approval Chamberlin--Courant based on consistency, highlighting principles of excellence, diversity, and proportionality.

## Contribution

It introduces counting rules and offers an axiomatic characterization of these rules based on the consistency axiom, clarifying their foundational principles.

## Key findings

- Characterizes counting rules through axioms.
- Provides axiomatic characterizations of three key multi-winner rules.
- Highlights principles of individual excellence, diversity, and proportionality.

## Abstract

This paper is an axiomatic study of consistent approval-based multi-winner rules, i.e., voting rules that select a fixed-size group of candidates based on approval ballots. We introduce the class of counting rules and provide an axiomatic characterization of this class based on the consistency axiom. Building upon this result, we axiomatically characterize three important consistent multi-winner rules: Proportional Approval Voting, Multi-Winner Approval Voting and the Approval Chamberlin--Courant rule. Our results demonstrate the variety of multi-winner rules and illustrate three different, orthogonal principles that multi-winner voting rules may represent: individual excellence, diversity, and proportionality.