Improved Metric Distortion Bounds for Deterministic Weighted-Tournament Voting Rules

Abstract

In metric social choice, voters and candidates lie in a common but unknown metric space, voters rank candidates by distance, and a voting rule seeks to minimize total distance to the voters. Its distortion is the worst-case approximation ratio relative to the minimum possible total distance. We study weighted-tournament rules (also known as C2 rules), which observe only the fraction of voters who prefer $a$ to $b$ for each pair of candidates $a,b$. These frequencies form a weighted tournament on candidates, a compressed representation that omits voter identities and the association of comparisons with individual voters. Prior work placed the optimal distortion of deterministic C2 rules between $3.1128$ and $3.9312$ [Charikar et al., EC 2025]. We introduce the Path-Unblanketed Set rule, a polynomial-time deterministic C2 rule with distortion at most $1+2\sqrt{2}\approx 3.8284$ for every finite number of candidates. For elections with no more than six candidates, we prove with computer assistance that the distortion is at most $3.3346$. Furthermore, using an exact computer-assisted certificate, we provide a lower bound of $3.1828$ for deterministic C2 rules as a byproduct.