We consider the following well-studied problem of metric distortion in social choice. Suppose we have an election with $n$ voters and $m$ candidates located in a shared metric space. We would like to design a voting rule that chooses a candidate whose average distance to the voters is small. However, instead of having direct access to the distances in the metric space, the voting rule obtains, from each voter, a ranked list of the candidates in order of distance. Can we design a rule that regardless of the election instance and underlying metric space, chooses a candidate whose cost differs from the true optimum by only a small factor (known as the distortion)? A long line of work culminated in finding optimal deterministic voting rules with metric distortion $3$. However, for randomized voting rules, there is still a gap in our understanding: Even though the best lower bound is $2.112$, the best upper bound is still $3$, attained even by simple rules such as Random Dictatorship. Finding a randomized rule that guarantees distortion $3 - \epsilon$ has been a major challenge in computational social choice, as prevalent approaches to designing voting rules are known to be insufficient. Such a voting rule must use information beyond aggregate comparisons between pairs of candidates, and cannot only assign positive probability to candidates that are voters' top choices. In this work, we give a rule that guarantees distortion less than $2.753$. To do so we study a handful of voting rules that are new to the problem. One is Maximal Lotteries, a rule based on the Nash equilibrium of a natural zero-sum game which dates back to the 60's. The others are novel rules that can be thought of as hybrids of Random Dictatorship and the Copeland rule. Though none of these rules can beat distortion $3$ alone, a randomization between Maximal Lotteries and any of the novel rules can.

翻译：我们考虑社会选择中一个被深入研究的度量失真问题。假设存在一场选举，其中n位选民和m位候选人位于一个共享的度量空间中。我们希望设计一种投票规则，使其选择的候选人与选民的平均距离较小。然而，投票规则无法直接获取度量空间中的距离信息，而只能从每位选民处获得按距离排序的候选人排名列表。我们能否设计一种规则，无论选举实例和底层度量空间如何，都能选择一个其代价与真实最优解仅相差较小因子（即失真度）的候选人？一系列长期研究最终找到了具有度量失真度3的最优确定性投票规则。然而，对于随机化投票规则，我们的理解仍存在差距：尽管最佳下界为2.112，但最佳上界仍为3，甚至简单规则如随机独裁也能达到该值。在计算社会选择领域，寻找能保证失真度3 - ε的随机化规则一直是一个重大挑战，因为已知的投票规则设计主流方法均存在不足。此类投票规则必须利用超越候选人对间聚合比较的信息，且不能仅将正概率分配给选民的首选候选人。在本研究中，我们提出了一种保证失真度小于2.753的规则。为此，我们研究了若干针对该问题新提出的投票规则：其一是最大彩票规则，该规则基于可追溯至60年代的自然零和博弈纳什均衡；其他则是可视为随机独裁与科普兰规则混合体的新型规则。尽管这些规则单独使用均无法突破失真度3，但最大彩票规则与任一新型规则的随机化组合能够实现这一突破。