We study the problem of designing voting rules that take as input the ordinal preferences of $n$ agents over a set of $m$ alternatives and output a single alternative, aiming to optimize the overall happiness of the agents. The input to the voting rule is each agent's ranking of the alternatives from most to least preferred, yet the agents have more refined (cardinal) preferences that capture the intensity with which they prefer one alternative over another. To quantify the extent to which voting rules can optimize over the cardinal preferences given access only to the ordinal ones, prior work has used the distortion measure, i.e., the worst-case approximation ratio between a voting rule's performance and the best performance achievable given the cardinal preferences. The work on the distortion of voting rules has been largely divided into two worlds: utilitarian distortion and metric distortion. In the former, the cardinal preferences of the agents correspond to general utilities and the goal is to maximize a normalized social welfare. In the latter, the agents' cardinal preferences correspond to costs given by distances in an underlying metric space and the goal is to minimize the (unnormalized) social cost. Several deterministic and randomized voting rules have been proposed and evaluated for each of these worlds separately, gradually improving the achievable distortion bounds, but none of the known voting rules perform well in both worlds simultaneously. In this work, we prove that one can achieve the best of both worlds by designing new voting rules, that simultaneously achieve near-optimal distortion guarantees in both distortion worlds. We also prove that this positive result does not generalize to the case where the voting rule is provided with the rankings of only the top-$t$ alternatives of each agent, for $t<m$.

翻译：[translated abstract in Chinese] 我们研究设计投票规则的问题，该规则以$n$个智能体对$m$个备选方案的序数偏好为输入，并输出单个备选方案，旨在优化智能体的总体满意度。投票规则的输入是每个智能体从最偏好到最不偏好的备选方案排序，然而智能体拥有更精细的（基数）偏好，这些偏好捕捉了它们对某一备选方案相对于另一备选方案的偏好强度。为了量化仅基于序数偏好进行优化的投票规则在基数偏好上的优化程度，先前工作使用了失真度量，即投票规则表现与给定基数偏好下可实现的最佳表现之间的最坏情况近似比。关于投票规则失真的研究主要分为两个世界：功利失真和度量失真。在前者中，智能体的基数偏好对应于一般效用，目标是最大化归一化的社会福利。在后者中，智能体的基数偏好对应于由潜在度量空间中距离给出的成本，目标是最小化（非归一化的）社会成本。已有若干确定性和随机性投票规则分别针对每个世界提出并评估，逐步改进了可达的失真界，但已知的投票规则中没有一种能同时在两个世界中表现良好。在本工作中，我们证明可以通过设计新的投票规则实现两全其美，这些规则在两个失真世界中同时达到接近最优的失真保证。我们还证明，当投票规则仅获得每个智能体的前$t$个备选方案排序（其中$t<m$）时，这一积极结果不会普遍化。