Metric Distortion of Obnoxious Distributed Voting
Alexandros A. Voudouris
TL;DR
This work analyzes obnoxious distributed metric voting, where agents and obnoxious alternatives lie in a metric space and decisions are made via two-step mechanisms that first select a group representative and then a final winner. It derives tight distortion bounds under two information regimes: full-information mechanisms achieve a bound of $2\min\{m,k\}-1$, with a line-metric special case reducing to 3, while ordinal mechanisms yield $3$ in centralized settings and $4\min\{m,k\}-1$ (with a line-refined 7) in distributed settings, all with matching lower bounds. The key technical tools include selecting group representatives via per-group optima in the full-information setting and using domination graphs and related structure in the ordinal setting. Overall, the results delineate the efficiency limits of two-step, information-limited mechanisms for obnoxious facility-like decisions and provide guidance for mechanism design under restricted information.
Abstract
We consider a distributed voting problem with a set of agents that are partitioned into disjoint groups and a set of obnoxious alternatives. Agents and alternatives are represented by points in a metric space. The goal is to compute the alternative that maximizes the total distance from all agents using a two-step mechanism which, given some information about the distances between agents and alternatives, first chooses a representative alternative for each group of agents, and then declares one of them as the overall winner. Due to the restricted nature of the mechanism and the potentially limited information it has to make its decision, it might not be always possible to choose the optimal alternative. We show tight bounds on the distortion of different mechanisms depending on the amount of the information they have access to; in particular, we study full-information and ordinal mechanisms.