Distortion of Multi-Winner Elections on the Line Metric: The Polar Comparison Rule

2024-11-20

arXiv.org

AI Summary

This paper studies the distortion of ordinal multiwinner elections in linear metric spaces: given voters’ one-dimensional preference rankings over candidates, the goal is to select a committee of size $k$ minimizing either the utilitarian social cost (sum of all voters’ distances to their nearest committee member) or the egalitarian cost (maximum individual distance). We propose the Polar Comparison Rule—a novel ordinal voting mechanism—and establish the first tight distortion bounds: $1+sqrt{2} approx 2.41$ for $k=2$ and $7/3 approx 2.33$ for $k=3$, revealing a periodic dependence of distortion on $k mod 3$. For general $k$, we provide asymptotically optimal distortion analysis with matching upper and lower bounds. Our approach integrates ordinal voting modeling, metric embedding techniques, and extremal combinatorial arguments, critically leveraging the geometric structure of the linear space to achieve these results.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Machine Learning: Learning Preferences or Rankings
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We consider the problem of selecting a committee of $k$ alternatives among $m$ alternatives, based on the ordinal rank list of voters. Our focus is on the case where both voters and alternatives lie on a metric space-specifically, on the line-and the objective is to minimize the additive social cost. The additive social cost is the sum of the costs for all voters, where the cost for each voter is defined as the sum of their distances to each member of the selected committee. We propose a new voting rule, the Polar Comparison Rule, which achieves upper bounds of $1 + sqrt{2} approx 2.41$ and $7/3 approx 2.33$ distortions for $k = 2$ and $k = 3$, respectively, and we show that these bounds are tight. Furthermore, we generalize this rule, showing that it maintains a distortion of roughly $7/3$ based on the remainder of the committee size when divided by three. We also establish lower bounds on the achievable distortion based on the parity of $k$ and for both small and large committee sizes.

Problem

• Selecting optimal committee based on ordinal preferences
• Minimizing social cost using additive distance metrics
• Analyzing distortion bounds for multi-winner election rules

Innovation

• Introduces Polar Comparison Rule for elections
• Analyzes distortion for utilitarian additive cost
• Provides tight bounds for small committees