发表机构
Saint Paul Academy and Summit School; Massachusetts Institute of Technology（圣保罗学院与顶峰学校; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无偏文化下 Smith 集合基数的渐近概率，证明其在不同基数范围下的衰减速率，并解决相关猜想。
AI 中文摘要
对于某个整数 $\ell\geq 2$，考虑 $m=2\ell-1$ 个选民，每个选民对 $n$ 个备选方案有一个偏好排序。Smith 集合是满足以下条件的最小非空备选方案集合：集合中的每个备选方案在成对多数比较中击败集合外的每个备选方案。在标准基准无偏文化下，即每个选民独立且均匀地随机选择对所有备选方案的偏好排序，我们研究当 $\ell$ 固定且 $n\to\infty$ 时，Smith 集合基数为 $s$ 的渐近概率。首先，我们证明 Smith 集合具有常数基数 $s$ 的概率以 $\Theta_{\ell,s} (n^{-s(\ell-1)/\ell})$ 的速率趋于零。其次，如果 $\min\{s,n-s\}\to\infty$，我们证明该概率以超多项式速率衰减。进一步，当 $s$ 和 $n-s$ 均为 $\Theta(n)$ 时，我们证明该概率以指数速率衰减，并确定精确的指数基数。最后，我们证明 Smith 集合包含所有备选方案的概率以 $\Theta_\ell(n^{-(\ell-1)/\ell})$ 的速率趋近于 1。这些定理解决了 [Liu et al, Ann Stat 2026] 在无偏文化下的猜想。
英文摘要
For some integer $\ell\geq 2$, consider $m=2\ell-1$ voters, each of which has a preference ranking over $n$ alternatives. The Smith set is the smallest nonempty set of alternatives each of which defeats every alternative outside the set in a pairwise majority comparison. Under the standard benchmark impartial culture where each voter uniformly and independently chooses a random preference ranking over all alternatives, we study the asymptotic probability that the Smith set has cardinality $s$ for fixed $\ell$ and $n\to \infty$. First, we prove that the probability that the Smith set has constant cardinality $s$ goes to zero at a rate of $Θ_{\ell,s} (n^{-s(\ell-1)/\ell})$. Next, if $\min\{s,n-s\}\to\infty$, we show that this probability decays superpolynomially. Further, when $s$ and $n-s$ are both $Θ(n)$, we prove that the probability decays exponentially and determine the exact exponential base. Finally, we prove that the probability that the Smith set contains all alternatives approaches one at a rate of $Θ_\ell(n^{-(\ell-1)/\ell})$. These theorems resolve the conjectures in [Liu et al, Ann Stat 2026] under impartial culture.
Comments20 pages
