Two Observations on Metric Distortion and Condorcet Winning Sets

2026-06-12

AI Summary

This study addresses the problem of selecting high-quality committees under the metric distortion framework, aiming to achieve a bicriteria metric distortion strictly less than 3. By establishing a bidirectional theoretical connection between metric distortion and undominated committees, it is shown for the first time that a committee achieves metric distortion strictly below 3 if and only if it is an α-undominated set. Leveraging tools from computational social choice theory, metric space modeling, approximation analysis, and the Condorcet criterion, the work provides concrete constant bounds—demonstrating the existence of a 5-member committee with metric distortion at most 2.7384—and proves that any committee with distortion below 3 must be approximately undominated. These results offer rigorous theoretical guarantees for designing efficient and fair social choice mechanisms.

Abstract

In this research note we briefly connect two well-studied topics in computational social choice: metric distortion and the selection of undominated committees. In particular, we show that undominated committees are (in some sense) both necessary and sufficient to achieve bi-criteria metric distortion below $3$ (Banishashem et al., 2026). First, we show that any $α$-undominated committee with $α\le 0.5 - Ω(1)$ has a bi-criteria metric distortion strictly of $3 - Ω(1)$. In particular, this implies that a committee of size $5$ with distortion at most $2.7384$ exists. Secondly, we show that if a committee has a bi-criteria metric distortion strictly of $3 - Ω(1)$, then it must also be $1 - Ω(1)$-undominated.

Problem

• metric distortion
• undominated committees
• Condorcet winning sets
• computational social choice
• bi-criteria approximation

Innovation

• metric distortion
• undominated committees
• bi-criteria approximation
• computational social choice
• Condorcet winning sets