确定性加权锦标赛投票规则的改进度量失真边界
Improved Metric Distortion Bounds for Deterministic Weighted-Tournament Voting Rules
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中文总结 AI 辅助
本文针对确定性加权锦标赛（C2）投票规则，提出路径未覆盖集规则，将其通用失真度上界优化至约3.8284，6候选者时上界为3.3346，同时给出其失真度下界3.1828。
中文摘要 AI 辅助
在度量社会选择中，选民与候选者处于共同但未知的度量空间内，选民依据距离对候选者排序，投票规则旨在最小化到选民的总距离，其失真度是相对于最小可能总距离的最坏情况近似比。本文研究加权锦标赛规则（又称C2规则），该规则仅观测每对候选者a、b中偏好a的选民比例，这些频率构成候选者上的加权锦标赛，是省略了选民身份及比较与单个选民关联的压缩表示。此前研究将确定性C2规则的最优失真度界定在3.1128至3.9312之间[Charikar等人，EC 2025]。本文提出路径未覆盖集规则，这是一种多项式时间确定性C2规则，对任意有限数量的候选者，其失真度至多为1+2√2≈3.8284；对于候选者不超过6个的选举，本文借助计算机辅助证明其失真度至多为3.3346；此外，作为副产品，本文利用精确的计算机辅助证书，给出确定性C2规则的下界为3.1828。
英文摘要
In metric social choice, voters and candidates lie in a common but unknown metric space, voters rank candidates by distance, and a voting rule seeks to minimize total distance to the voters. Its distortion is the worst-case approximation ratio relative to the minimum possible total distance. We study weighted-tournament rules (also known as C2 rules), which observe only the fraction of voters who prefer $a$ to $b$ for each pair of candidates $a,b$. These frequencies form a weighted tournament on candidates, a compressed representation that omits voter identities and the association of comparisons with individual voters. Prior work placed the optimal distortion of deterministic C2 rules between $3.1128$ and $3.9312$ [Charikar et al., EC 2025]. We introduce the Path-Unblanketed Set rule, a polynomial-time deterministic C2 rule with distortion at most $1+2\sqrt{2}\approx3.8284$ for every finite number of candidates. For elections with no more than six candidates, we prove with computer assistance that the distortion is at most $3.3346$. Furthermore, using an exact computer-assisted certificate, we provide a lower bound of $3.1828$ for deterministic C2 rules as a byproduct.
