  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.
