We consider a two-round election model involving $m$ voters and $n$ candidates. Each voter is endowed with a strict preference list ranking the candidates. In the first round, the candidates are partitioned into two subsets, $A$ and $B$, and voters select their preferred candidate from each. Provided there are no ties, the two respective winners advance to a second round, where voters choose between them according to their initial preference lists. We analyze this scenario using a probabilistic framework based on a spatial voting model with cyclically constructed preference lists and uniformly distributed ideal points. Our objective is to determine the optimal initial partition of $A$ and $B$ that maximizes a target candidate's probability of winning. We analytically evaluate this success probability and derive its asymptotic behavior as the number of candidates $n \to \infty$. A key finding is that the asymptotically optimal relative width of the main discrete cluster converges precisely to one-fifth of the total number of candidates. Finally, we provide computational results and confidence intervals derived from simulation algorithms that validate the analytical framework. Specifically, we demonstrate that the probability of the universal victory event rapidly approaches $1$ as the electorate size increases.

翻译：本文研究一个包含$m$位选民与$n$位候选人的两轮选举模型。每位选民被赋予一个对候选人进行严格排序的偏好列表。在第一轮中，候选人被划分为两个子集$A$和$B$，选民从每个子集中选择其最偏好的候选人。若无平局，则两个子集的胜出者进入第二轮，选民根据其初始偏好列表在两者之间进行选择。我们基于空间投票模型，采用循环构造的偏好列表与均匀分布的理想点，通过概率框架分析这一场景。研究目标是确定能够最大化目标候选人胜选概率的最优初始分区$A$和$B$。我们通过解析方法计算该成功概率，并推导当候选人数量$n \to \infty$时的渐近行为。一个关键发现是：主离散簇的最优相对宽度渐近收敛于候选人总数的五分之一。最后，我们通过仿真算法给出计算结果与置信区间，验证了分析框架的有效性。具体而言，我们证明随着选民规模增大，全局胜利事件的概率迅速趋近于$1$。