Dimensionality, Coordination, and Robustness in Voting

TL;DR

This paper analyzes voting mechanisms' performance using metric-distortion, revealing how intrinsic dimensionality affects outcomes, and introduces simple learning rules that achieve low distortion, with implications for facility location and low-dimensional spaces.

Contribution

It establishes a link between voting rule performance and metric space dimensionality, provides bounds on STV distortion, and introduces decentralized learning dynamics for low-distortion outcomes.

Findings

STV distortion is $O(d \, \log \log m)$ in doubling metric spaces.

Decentralized exploration/exploitation dynamics converge to candidates with $O(1)$ distortion.

Mechanisms like Gkatzelis et al.'s attain optimal distortion of 2 in ultra-metrics.

Abstract

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.