Sensitivity and Differential Privacy in Metric Voting with Distortion below Three

Abstract

Voting rules aggregate individual preferences into collective decisions, but the rankings they receive contain only ordinal information. The metric distortion framework studies ordinal voting rules in settings where voters and candidates are embedded in an unknown metric space. Deterministic rules have optimal worst-case distortion $3$, while recent randomized rules break the $3$ barrier. We study whether such improvements can coexist with low worst-case sensitivity with respect to the Wasserstein distance of lotteries under one-voter deletion and approximate differential privacy under one-voter replacement. On the sensitivity side, we give a randomized rule with distortion at most $3-\varepsilon$ for an absolute constant $\varepsilon>0$ and, for $m$ candidates and $n$ voters, a worst-case sensitivity bound of $O((\log m+1)/n)$. On the privacy side, for every $\delta\in(0,1)$ and all $n$ above an absolute constant, we construct a variant rule whose mechanism releasing a single sampled winner has distortion at most $3-\varepsilon$ and is $(O((\log m+\log(1/\delta)+1)/n),\delta)$-differentially private. Both constructions use the same family of Gibbs distributions over constant-size candidate lists, with only the temperature parameter differing between the sensitivity and differential-privacy guarantees. Our analysis builds on the biased-metric viewpoint behind the recent improvement over the $3$ barrier and proves a stability property for the biased-metric ratio.