The Metric Distortion of Randomized Social Choice Functions: C1 Maximal Lottery Rules and Simulations

The metric distortion of a randomized social choice function (RSCF) quantifies its worst-case approximation ratio of the optimal social cost when the voters' costs for alternatives are given by distances in a metric space. This notion has recently attracted significant attention as numerous RSCFs that aim to minimize the metric distortion have been suggested. However, such tailored voting rules usually have little appeal other than their low metric distortion. In this paper, we will thus study the metric distortion of well-established RSCFs. In more detail, we first show that C1 maximal lottery rules, a well-known class of RSCFs, have a metric distortion of $4$ and furthermore prove that this is optimal within the class of majoritarian RSCFs (which only depend on the majority relation). As our second contribution, we perform extensive computer experiments on the metric distortion of established RSCFs to obtain insights into their average-case performance. These computer experiments are based on a new linear program for computing the metric distortion of a lottery on a given profile and reveal that some classical RSCFs perform almost as well as the currently best known RSCF with respect to the metric distortion on randomly sampled profiles.