We study social choice rules under the utilitarian distortion framework, with an additional metric assumption on the agents' costs over the alternatives. In this approach, these costs are given by an underlying metric on the set of all agents plus alternatives. Social choice rules have access to only the ordinal preferences of agents but not the latent cardinal costs that induce them. Distortion is then defined as the ratio between the social cost (typically the sum of agent costs) of the alternative chosen by the mechanism at hand, and that of the optimal alternative. This model was introduced by Anshelevich et al. (2015) who conjectured that Ranked Pairs, the well-known social choice rule, achieves a distortion of at most 3. We disprove this conjecture by constructing a sequence of instances which shows that the worst-case distortion of Ranked Pairs is at least 5. Our lower bound on the worst case distortion of Ranked Pairs matches a previously known upper bound for the Copeland rule, proving that in the worst case, the (simpler) Copeland rule is at least as good as Ranked Pairs. And as long as we are limited to \emph{tournament} rules, we demonstrate that randomization cannot help achieve an expected worst-case distortion of less than 3. Using the concept of approximate majorization within the distortion framework, we prove that Copeland and Randomized Dictatorship achieve low constant factor fairness-ratios (5 and 3 respectively); this approximates essentially all fairness measures, and is a considerable generalization of the work of Anshelevich et al. (2015) who prove similar results for objectives such as the sum and median of agent costs. In addition to all of the above, we outline several interesting directions for further research in this space.
['Ashish Goel', 'Anilesh Kollagunta Krishnaswamy', 'Kamesh Munagala']
Metric Distortion of Social Choice Rules: Lower Bounds and Fairness Properties