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.

翻译：随机社会选择函数的度量失真量化了当选民对备选方案的成本由度量空间中的距离给出时，其在最坏情况下对最优社会成本的逼近比率。该概念近期吸引了大量关注，因为已提出众多旨在最小化度量失真的随机社会选择函数。然而，这类定制化投票规则除低度量失真外通常缺乏其他吸引力。因此，本文将研究已有成熟随机社会选择函数的度量失真。具体而言，我们首先证明C1最大抽签规则——一类著名的随机社会选择函数——其度量失真为$4$，并进一步证明这在多数决型随机社会选择函数（仅依赖于多数关系）类中是最优的。作为第二项贡献，我们对现有随机社会选择函数的度量失真进行大规模计算机实验，以洞悉其平均表现。这些计算机实验基于一个用于计算给定偏好分布下抽签度量失真的新线性规划，结果表明某些经典随机社会选择函数在随机采样分布上的度量失真表现几乎与当前已知最优的随机社会选择函数相当。