The Condorcet Dimension of Metric Spaces

TL;DR

This paper investigates the Condorcet dimension in spatial voting models, establishing that in two-dimensional proximity-based elections, the minimum size of a Condorcet winning set is at most three, under common distance norms.

Contribution

It proves that the Condorcet dimension is at most three in 2D metric spaces with proximity preferences, extending understanding of voting stability in spatial models.

Findings

Condorcet dimension is at most 3 in 2D proximity elections.

Results hold under Manhattan and infinity norms.

Advances theoretical understanding of voting stability in spatial settings.

Abstract

A Condorcet winning set is a set of candidates such that no other candidate is preferred by at least half the voters over all members of the set. The Condorcet dimension, which is the minimum cardinality of a Condorcet winning set, is known to be at most logarithmic in the number of candidates. We study the case of elections where voters and candidates are located in a $2$-dimensional space with preferences based upon proximity voting. Our main result is that the Condorcet dimension is at most $3$, under both the Manhattan norm and the infinity norm, natural measures in electoral systems.