Metric-Distortion Bounds under Limited Information

Abstract

In this work, we study the metric distortion problem in voting theory under a limited amount of ordinal information. Our primary contribution is threefold. First, we consider mechanisms that perform a sequence of pairwise comparisons between candidates. We show that a popular deterministic mechanism employed in many knockout phases yields distortion O(log m) while eliciting only m − 1 out of the Θ(m² ) possible pairwise comparisons, where m represents the number of candidates. Our analysis for this mechanism leverages a powerful technical lemma developed by Kempe (AAAI ‘20). We also provide a matching lower bound on its distortion. In contrast, we prove that any mechanism which performs fewer than m−1 pairwise comparisons is destined to have unbounded distortion. Moreover, we study the power of deterministic mechanisms under incomplete rankings. Most notably, when agents provide their k-top preferences we show an upper bound of 6m/k + 1 on the distortion, for any k ∈ {1, 2, . . . , m}. Thus, we substantially improve over the previous bound of 12m/k established by Kempe (AAAI ‘20), and we come closer to matching the best-known lower bound. Finally, we are concerned with the sample complexity required to ensure near-optimal distortion with high probability. Our main contribution is to show that a random sample of Θ(m/ϵ² ) voters suffices to guarantee distortion 3 + ϵ with high probability, for any sufficiently small ϵ > 0. This result is based on analyzing the sensitivity of the deterministic mechanism introduced by Gkatzelis, Halpern, and Shah (FOCS ‘20). Importantly, all of our sample-complexity bounds are distribution-independent.

From an experimental standpoint, we present several empirical findings on real-life voting applications, comparing the scoring systems employed in practice with a mechanism explicitly minimizing (metric) distortion. Interestingly, for our case studies, we find that the winner in the actual competition is typically the candidate who minimizes the distortion.