带界随机性下的度量投票的鲁棒彩票压缩：一种转移原理
Robust Lottery Compression for Metric Voting: A Transfer Principle for Bounded Randomness
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中文总结 AI 辅助
该研究针对带界随机性下的度量投票，提出维度无关的彩票压缩定理，证明K≥802时规则失真度可接近5/2，164个条目即可突破确定性失真3的障碍，收敛速率为O(K^(-1/3))。
中文摘要 AI 辅助
我们研究带界随机性下随机社会选择中的度量失真：在每个偏好配置上，投票规则必须确定性地确定K个候选人的多重集，然后从中均匀随机选取一个。此前研究表明，这种受限模型的表现优于最优确定性失真值3。我们证明其实际上可以接近当前无约束的最佳上界5/2。对于每个整数K≥802，存在一个带界随机性规则，其失真度至多为5/2 +3(π/(8K))^(1/3) +2√(π/(8K))。因此，对于失真度5/2+ε，仅需O(ε^(-3))个条目即可，且与选民和候选人的数量无关。我们还证明，164个条目已能实现严格低于3的失真度，这给出了突破确定性障碍所需的最小列表大小2≤N*≤164。我们的主要技术贡献是一个与维度无关的压缩定理：若某彩票的失真度至多为ρ，且其支撑集中的每个候选人的确定性失真度至多为H，则它存在一个均匀K个条目的近似，其失真度至多为ρ+(H+1)√(π/(8K))。因此，可能表现良好的彩票仅会产生O(K^(-1/2))的压缩损失。混合集成否决（Mixed Integrated Veto）不满足该支撑条件，故我们先移除提前被淘汰的结果，以O(τ²)的失真度损失换取每个受支撑候选人的确定性失真度的O(1/τ)界。平衡该修复成本与压缩成本，即可得到O(K^(-1/3))的收敛速率。
英文摘要
We study metric distortion in randomized social choice under bounded randomness: on every preference profile, the voting rule must deterministically identify a multiset of $K$ candidates and then select a uniformly random entry. Previous work showed that this restricted model can beat the optimal deterministic distortion of $3$. We show that it can in fact approach the current best unrestricted upper benchmark of $5/2$. For every integer $K\ge 802$, there exists a bounded-randomness rule with distortion at most $\frac{5}{2} +3\left(\fracπ{8K}\right)^{1/3} +2\sqrt{\fracπ{8K}}$. Consequently, $O(\varepsilon^{-3})$ entries suffice for distortion $5/2+\varepsilon$, independently of the numbers of voters and candidates. We also show that $164$ entries already achieve distortion strictly below $3$, giving $2\le N^\star\le 164$ for the minimum list size needed to break the deterministic barrier. Our main technical contribution is a dimension-free compression theorem: if a lottery has distortion at most $ρ$ and every candidate in its support has deterministic distortion at most $H$, then it admits a uniform $K$-entry approximation with distortion at most $ρ+(H+1)\sqrt{π/(8K)}$. Thus, lotteries whose possible outcomes are already well behaved incur only $O(K^{-1/2})$ compression loss. Mixed Integrated Veto does not satisfy this support condition, so we first remove early-eliminated outcomes, trading $O(τ^2)$ distortion loss for an $O(1/τ)$ bound on the deterministic distortion of every supported candidate. Balancing this repair cost against compression yields the $O(K^{-1/3})$ convergence rate.
