The Condorcet Dimension of Metric Spaces

TL;DR

The paper investigates the Condorcet dimension within a 2D spatial voting framework using proximity-based preferences under the Manhattan and ∞-norms. It proves an upper bound of 3 on the Condorcet dimension for these norms by a four-quadrant construction and a refinement showing a 3-candidate Condorcet winning set exists, and then links these results to a broader embedding capability: any preference profile can be embedded in D = min[m,n] dimensions for any p-norm in polynomial time. A simple 2D lower-bound demonstrates Condorcet dimension can be at least 2 for any p-norm, and the paper also discusses a rotation argument to translate ∞-norm results from Manhattan. Together, these results deepen the understanding of how spatial constraints influence Condorcet dynamics and provide practical embedding techniques for arbitrary profiles in metric spaces.

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.

Figures (1)

• Figure 2: Using the Manhattan norm in (a), we have $d^1({\bf 0}, {\bf p}_1)=4, d^1({\bf 0}, {\bf p}_2)=2$ and $d^1({\bf p_1}, {\bf p}_2)=6$. Rotating the axis by 45 degrees and the applying the infinity norm gives $\hat{d}^\infty({\bf 0}, {\bf p}_1)= 2\sqrt{2}= \frac{1}{\sqrt{2}}\cdot 4, \hat{d}^\infty({\bf 0}, {\bf p}_2)=\sqrt{2}=\frac{1}{\sqrt{2}}\cdot 2$ and $\hat{d}^\infty({\bf p_1}, {\bf p}_2)= 3\sqrt{2}=\frac{1}{\sqrt{2}}\cdot 6$.

Theorems & Definitions (18)

• Theorem 1: ELS15
• proof
• Lemma 2
• proof
• Corollary 3
• proof
• Theorem 4
• proof
• Claim 5
• proof