  We study the performance of voting mechanisms from a utilitarian standpoint,
under the recently introduced framework of metric-distortion, offering new
insights along three main lines. First, if $d$ represents the doubling
dimension of the metric space, we show that the distortion of STV is $O(d \log
\log m)$, where $m$ represents the number of candidates. For doubling metrics
this implies an exponential improvement over the lower bound for general
metrics, and as a special case it effectively answers a question left open by
Skowron and Elkind (AAAI '17) regarding the distortion of STV under
low-dimensional Euclidean spaces. More broadly, this constitutes the first
nexus between the performance of any voting rule and the "intrinsic
dimensionality" of the underlying metric space. We also establish a
nearly-matching lower bound, refining the construction of Skowron and Elkind.
Moreover, motivated by the efficiency of STV, we investigate whether natural
learning rules can lead to low-distortion outcomes. Specifically, we introduce
simple, deterministic and decentralized exploration/exploitation dynamics, and
we show that they converge to a candidate with $O(1)$ distortion. Finally,
driven by applications in facility location games, we consider several
refinements and extensions of the standard metric-setting. Namely, we prove
that the deterministic mechanism recently introduced by Gkatzelis, Halpern, and
Shah (FOCS '20) attains the optimal distortion bound of $2$ under
ultra-metrics, while it also comes close to our lower bound under distances
satisfying approximate triangle inequalities.
