The metric distortion framework posits that n voters and m candidates are jointly embedded in a metric space such that voters rank candidates that are closer to them higher. A voting rule's purpose is to pick a candidate with minimum total distance to the voters, given only the rankings, but not the actual distances. As a result, in the worst case, each deterministic rule picks a candidate whose total distance is at least three times larger than that of an optimal one, i.e., has distortion at least 3. A recent breakthrough result showed that achieving this bound of 3 is possible; however, the proof is non-constructive, and the voting rule itself is a complicated exhaustive search. Our main result is an extremely simple voting rule, called Plurality Veto, which achieves the same optimal distortion of 3. Each candidate starts with a score equal to his number of first-place votes. These scores are then gradually decreased via an n-round veto process in which a candidate drops out when his score reaches zero. One after the other, voters decrement the score of their bottom choice among the standing candidates, and the last standing candidate wins. We give a one-paragraph proof that this voting rule achieves distortion 3. This rule is also immensely practical, and it only makes two queries to each voter, so it has low communication overhead. We also generalize Plurality Veto into a class of randomized voting rules in the following way: Plurality veto is run only for k < n rounds; then, a candidate is chosen with probability proportional to his residual score. This general rule interpolates between Random Dictatorship (for k=0) and Plurality Veto (for k=n-1), and k controls the variance of the output. We show that for all k, this rule has distortion at most 3.

翻译：度量失真框架假设n个选民和m个候选人共同嵌入一个度量空间，使得选民对距离更近的候选人给予更高排名。投票规则的目的是仅根据排名（而非实际距离）选出与选民总距离最小的候选人。最坏情况下，每个确定性规则选出的候选人总距离至少是最优候选人的三倍（即失真至少为3）。近期一项突破性结果表明，实现这一3倍界限是可能的；然而，该证明是非构造性的，且投票规则本身是一种复杂的穷举搜索。我们的主要结果是提出一种极其简单的投票规则——多数否决，它同样达到了3的最优失真。每位候选人初始得分为其获得的第一选择票数，随后通过一个n轮否决过程逐步扣减分数：当候选人得分降至零时被淘汰。选民依次扣减当前在选候选人中自己最不赞成者的得分，最后存活的候选人获胜。我们仅用一段文字证明该投票规则失真为3。该规则极为实用，且每位选民仅需回答两次查询，因此通信开销极低。我们还将多数否决推广为一类随机投票规则：仅运行k<n轮多数否决；随后以与剩余分数成比例的概率选择候选人。该通用规则在随机独裁规则（k=0）与多数否决规则（k=n-1）之间插值，且k控制输出的方差。我们证明对所有k，该规则失真均不超过3。