AI 中文总结
该研究在度量投票中，构造了突破失真3界限的随机规则，同时实现了低最坏情况敏感性与差分隐私，为序数投票规则的隐私与效率平衡提供了新方案。
AI 中文摘要
投票规则将个人偏好汇总为集体决策，但它们获得的排名仅包含序数信息。度量失真框架研究选民和候选人嵌入未知度量空间环境中的序数投票规则，确定性规则的最坏情况失真为3，而近期的随机规则突破了3的界限。我们研究此类改进是否能与低最坏情况敏感性（关于单选民删除下的彩票Wasserstein距离）、以及单选民替换下的近似差分隐私共存。在敏感性方面，我们给出了一个随机规则，其失真最多为3−ε（ε>0为绝对常数），对于m个候选人、n个选民，最坏情况敏感性边界为O((log m + 1)/n)。在隐私方面，对于每个δ∈(0,1)和所有高于绝对常数的n，我们构造了一个变体规则，其机制释放单个抽样获胜者，失真最多为3−ε，且满足(O((log m + log(1/δ) + 1)/n),δ)-差分隐私。两种构造均使用同一族常数大小候选人列表上的Gibbs分布，仅温度参数在敏感性和差分隐私保证间存在差异。我们的分析基于近期突破3界限背后的偏置度量视角，并证明了偏置度量比率的稳定性属性。
英文摘要
Voting rules aggregate individual preferences into collective decisions, but the rankings they receive contain only ordinal information. The metric distortion framework studies ordinal voting rules in settings where voters and candidates are embedded in an unknown metric space. Deterministic rules have optimal worst-case distortion $3$, while recent randomized rules break the $3$ barrier. We study whether such improvements can coexist with low worst-case sensitivity with respect to the Wasserstein distance of lotteries under one-voter deletion and approximate differential privacy under one-voter replacement. On the sensitivity side, we give a randomized rule with distortion at most $3-\varepsilon$ for an absolute constant $\varepsilon>0$ and, for $m$ candidates and $n$ voters, a worst-case sensitivity bound of $O((\log m+1)/n)$. On the privacy side, for every $δ\in(0,1)$ and all $n$ above an absolute constant, we construct a variant rule whose mechanism releasing a single sampled winner has distortion at most $3-\varepsilon$ and is $(O((\log m+\log(1/δ)+1)/n),δ)$-differentially private. Both constructions use the same family of Gibbs distributions over constant-size candidate lists, with only the temperature parameter differing between the sensitivity and differential-privacy guarantees. Our analysis builds on the biased-metric viewpoint behind the recent improvement over the $3$ barrier and proves a stability property for the biased-metric ratio.
