We study the design of voting rules in the metric distortion framework. It is known that any deterministic rule suffers distortion of at least $3$, and that randomized rules can achieve distortion strictly less than $3$, often at the cost of reduced transparency and interpretability. In this work, we explore the trade-off between these paradigms by asking whether it is possible to break the distortion barrier of $3$ using only "bounded" randomness. We answer in the affirmative by presenting a voting rule that (1) achieves distortion of at most $3 - \varepsilon$ for some absolute constant $\varepsilon > 0$, and (2) selects a winner uniformly at random from a deterministically identified list of constant size. Our analysis builds on new structural results for the distortion and approximation of Maximal Lotteries and Stable Lotteries.

翻译：我们研究了度量失真框架下投票规则的设计。已知任何确定性规则的失真度至少为$3$，而随机化规则可以实现严格小于$3$的失真度，但往往以降低透明度和可解释性为代价。在这项工作中，我们通过探讨是否仅使用"有限"随机性即可突破$3$的失真壁垒，来研究这两种范式之间的权衡。我们给出了肯定回答，提出了一种投票规则，该规则（1）对于某个绝对常数$\varepsilon > 0$，实现至多$3 - \varepsilon$的失真度，且（2）从一个确定性确定的常数大小列表中均匀随机选择获胜者。我们的分析基于对最大彩票和稳定彩票的失真度与近似性的新结构结果。