Favorite-Candidate Voting for Eliminating the Least Popular Candidate in Metric Spaces

TL;DR

This paper investigates mechanisms for eliminating the least popular candidate in metric space voting, aiming to optimize social value and analyzing their worst-case performance bounds.

Contribution

It introduces new mechanisms for candidate elimination in metric spaces and establishes bounds on their distortion compared to optimal social value.

Findings

Derived upper bounds on mechanism distortion in general metrics.

Established lower bounds showing limitations of certain mechanisms.

Analyzed special cases with improved distortion bounds.

Abstract

We study single-candidate voting embedded in a metric space, where both voters and candidates are points in the space, and the distances between voters and candidates specify the voters' preferences over candidates. In the voting, each voter is asked to submit her favorite candidate. Given the collection of favorite candidates, a mechanism for eliminating the least popular candidate finds a committee containing all candidates but the one to be eliminated. Each committee is associated with a social value that is the sum of the costs (utilities) it imposes (provides) to the voters. We design mechanisms for finding a committee to optimize the social value. We measure the quality of a mechanism by its distortion, defined as the worst-case ratio between the social value of the committee found by the mechanism and the optimal one. We establish new upper and lower bounds on the distortion of mechanisms in this single-candidate voting, for both general metrics and well-motivated special cases.