We consider the following well studied problem of metric distortion in social choice. Suppose we have an election with $n$ voters and $m$ candidates who lie 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, each voter gives us 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 deterministic voting rules with metric distortion $3$, which is the best possible for deterministic rules and many other classes of voting rules. However, without any restrictions, there is still a significant gap in our understanding: Even though the best lower bound is substantially lower at $2.112$, the best upper bound is still $3$, which is attained even by simple rules such as Random Dictatorship. Finding a rule that guarantees distortion $3 - \varepsilon$ for some constant $\varepsilon $ has been a major challenge in computational social choice. 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 careful randomization between Maximal Lotteries and any of the novel rules can.

翻译：我们考虑社会选择中以下众所周知的度量失真问题。假设有一个包含$n$个选民和$m$个候选人的选举，他们位于一个共享的度量空间中。我们希望设计一种投票规则，使得选出的候选人与选民的平均距离较小。然而，选民无法直接获取度量空间中的距离，而是提供一份按距离排序的候选人名单。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，都能选出一个成本与真正最优解仅相差一个小因子（称为失真）的候选人？一系列研究工作最终发现了失真为$3$的确定性投票规则，这是确定性规则及许多其他类别投票规则所能达到的最佳结果。然而，在不加任何限制的情况下，我们的理解仍存在显著差距：尽管最佳下界明显更低（为$2.112$），但最佳上界仍为$3$，甚至简单的规则（如随机独裁）也能达到这一值。寻找一种能保证失真小于某个常数$\varepsilon$（即$3 - \varepsilon$）的规则，一直是计算社会选择中的一个重大挑战。在本文中，我们给出了一种保证失真小于$2.753$的规则。为此，我们研究了几种对该问题而言全新的投票规则。一种是最大抽签，这是一种基于自然零和博弈纳什均衡的规则，可追溯到20世纪60年代。其他规则则是新颖的规则，可以被视为随机独裁与科普兰规则的混合体。尽管这些规则单独使用时均无法打破失真$3$的界限，但将最大抽签与任意一种新规则进行精心随机化组合，便可实现这一目标。