We study the committee selection problem in the canonical impartial culture model with a large number of voters and an even larger candidate set. Here, each voter independently reports a uniformly random preference order over the candidates. For a fixed committee size $k$, we ask when a committee can collectively beat every candidate outside the committee by a prescribed majority level $α$. We focus on two natural notions of collective dominance, $α$-winning and $α$-dominating sets, and we identify sharp threshold phenomena for both of them using probabilistic methods, duality arguments, and rounding techniques. We first consider $α$-winning sets. A set $S$ of $k$ candidates is $α$-winning if, for every outside candidate $a \notin S$, at least an $α$-fraction of voters rank some member of $S$ above $a$. We show a sharp threshold at \[ α_{\mathrm{win}}^\star = 1 - \frac{1}{k}. \] Specifically, an $α$-winning set of size $k$ exists with high probability when $α< α_{\mathrm{win}}^\star$, and is unlikely to exist when $α> α_{\mathrm{win}}^\star$. We then study the stronger notion of $α$-dominating sets. A set $S$ of $k$ candidates is $α$-dominating if, for every outside candidate $a \notin S$, there exists a single committee member $b \in S$ such that at least an $α$-fraction of voters prefer $b$ to $a$. Here we establish an analogous sharp threshold at \[ α_{\mathrm{dom}}^\star = \frac{1}{2} - \frac{1}{2k}. \] As a corollary, our analysis yields an impossibility result for $α$-dominating sets: for every $k$ and every $α> α_{\mathrm{dom}}^\star = 1 / 2 - 1 / (2k)$, there exist preference profiles that admit no $α$-dominating set of size $k$. This corollary improves the best previously known bounds for all $k \geq 2$.

翻译：我们在典型中立文化模型中研究委员会选择问题，该模型包含大量选民和更大规模的候选人集合。在此模型中，每位选民独立地报告对候选人的均匀随机偏好排序。对于固定的委员会规模$k$，我们探讨委员会何时能够以预设的多数水平$α$集体击败委员会外的每位候选人。我们聚焦于两种自然的集体支配概念——$α$-获胜集与$α$-支配集，并运用概率方法、对偶论证和取整技术为两者识别出尖锐的阈值现象。首先考虑$α$-获胜集：若对于每个外部候选人$a \notin S$，至少有$α$比例的选民将$S$中某成员排在$a$之前，则$k$个候选人的集合$S$称为$α$-获胜集。我们证明在\[ α_{\mathrm{win}}^\star = 1 - \frac{1}{k} \]处存在尖锐阈值：当$α< α_{\mathrm{win}}^\star$时，规模为$k$的$α$-获胜集高概率存在；当$α> α_{\mathrm{win}}^\star$时则几乎不可能存在。随后研究更强的$α$-支配集概念：若对于每个外部候选人$a \notin S$，存在单个委员会成员$b \in S$使得至少有$α$比例的选民更偏好$b$而非$a$，则$k$个候选人的集合$S$称为$α$-支配集。在此我们建立了类似的尖锐阈值\[ α_{\mathrm{dom}}^\star = \frac{1}{2} - \frac{1}{2k} \]。作为推论，我们的分析给出了$α$-支配集的不可能性结果：对于任意$k$及任意$α> α_{\mathrm{dom}}^\star = 1 / 2 - 1 / (2k)$，总存在不包含任何规模为$k$的$α$-支配集的偏好剖面。该推论对所有$k \geq 2$的情形改进了先前已知的最佳界限。