Multi-Winner Voting with Approval Preferences (2007.01795v5)

Abstract: Multi-winner voting is the process of selecting a fixed-size set of representative candidates based on voters' preferences. It occurs in applications ranging from politics (parliamentary elections) to the design of modern computer applications (collaborative filtering, dynamic Q&A platforms, diversifying search results). All these applications share the problem of identifying a representative subset of alternatives -- and the study of multi-winner voting is the principled analysis of this task. This book provides a thorough and in-depth look at multi-winner voting based on approval preferences. One speaks of approval preferences if voters express their preferences by providing a set of candidates they approve. Approval preferences thus separate candidates in approved and disapproved ones, a simple, binary classification. The corresponding multi-winner voting rules are called approval-based committee (ABC) rules. Due to the simplicity of approval preferences, ABC rules are widely suitable for practical use. Recent years have seen a rising interest in ABC voting. While multi-winner voting has been originally a topic studied by economists and political scientists, a significant share of recent progress has occurred in the field of computational social choice. This discipline is situated in the intersection of artificial intelligence, computer science, economics, and (to a lesser degree) political science, combining insights and methods from these distinct fields. The goal of this book is to present fundamental concepts and results for ABC voting and to discuss the recent advances in computational social choice. The main focus is on axiomatic analysis, algorithmic results, and relevant applications.

• The paper provides an overview and analysis of approval-based committee (ABC) voting rules, examining different methods like PAV and AV for selecting representative groups.
• It examines core ABC voting rules such as Proportional Approval Voting (PAV) and Approval Voting (AV), analyzing their axiomatic properties including proportionality and monotonicity.
• The work addresses computational challenges in implementing complex rules like PAV and highlights implications for achieving fairer representation in elections and algorithmic systems.

Multi-Winner Voting with Approval Preferences: An Overview

The paper "Multi-Winner Voting with Approval Preferences" by Martin Lackner and Piotr Skowron explores an essential aspect of collective decision-making: selecting a representative group of candidates from a larger set based on voters' preferences. This multi-winner voting system has widespread implications, from political elections to computational recommendations.

Core Concepts and Methods

At the heart of this work is the examination of approval-based committee (ABC) voting rules, where voters express preferences by approving or disapproving candidates. This binary preference system simplifies voters' input and is adaptable to numerous applications. The authors highlight several ABC voting rules, including notable ones like Approval Voting (AV), Proportional Approval Voting (PAV), Sequential PAV, and the Method of Equal Shares.

1. Approval Voting (AV): This rule selects the top k candidates with the most approvals. It is straightforward but can overlook minority interests.
2. Proportional Approval Voting (PAV): PAV aims to balance the representation by using harmonic weights, thus attempting to achieve proportional representation.
3. Sequential PAV and seq-e's Rule: These methods build upon PAV by incrementally selecting candidates to maximize the overall approval score, providing a feasible polynomial-time implementation that maintains proportional properties in practice.
4. Method of Equal Shares: A rule where voters have a budget and spend indiscriminately on candidates, ensuring proportional representation by allowing the coalition of like-minded voters to select candidates cooperatively.

Axiomatic Analysis

The paper explores key axioms and properties that these ABC rules must navigate:

• Proportionality: Ensuring various voter groups are adequately represented in the elected committee.
• Monotonicity: The system's behavior with increasing voter support for a candidate.
• Support Monotonicity: Addresses the impact of additional voter approvals, a critical aspect for practical elections where population changes occur.
• Strategyproofness: Evaluates whether voters have incentives to misrepresent their preferences to influence outcomes beneficially.

Computational Aspects

Several ABC voting rules are computationally challenging to compute due to their optimization nature. The paper addresses these challenges by exploring approximation algorithms, emphasizing the balance between computational feasibility and desired systemic properties.

Implications and Future Directions

This comprehensive exploration of ABC voting systems underscores their potential to bring fairer representation in multi-winner contexts. The authors indicate that while some methods (like AV) are computationally simple, they might not achieve proportionality. Others, like PAV and seq-e, provide a balance between complexity and fairness, yielding significant implications for electoral systems and algorithmic fairness in AI applications.

Future research directions include refining computational methods to address scalability in large elections and further exploring these systems' strategic aspects to ensure robustness against manipulation while preserving democratic ideals.

In summary, the work by Lackner and Skowron is pivotal in analyzing how voting systems can be designed to reflect collective preferences proportionally and efficiently, making it a cornerstone for both theoretical investigation and practical application.