In an election in which each voter ranks all of the candidates, we consider the head-to-head results between each pair of candidates and form a labeled directed graph, called the margin graph, which contains the margin of victory of each candidate over each of the other candidates. A central issue in developing voting methods is that there can be cycles in this graph, where candidate $\mathsf{A}$ defeats candidate $\mathsf{B}$, $\mathsf{B}$ defeats $\mathsf{C}$, and $\mathsf{C}$ defeats $\mathsf{A}$. In this paper we apply the central limit theorem, graph homology, and linear algebra to analyze how likely such situations are to occur for large numbers of voters. There is a large literature on analyzing the probability of having a majority winner; our analysis is more fine-grained. The result of our analysis is that in elections with the number of voters going to infinity, margin graphs that are more cyclic in a certain precise sense are less likely to occur.

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大量选民随机选举的分析

在每个选民对所有候选人进行排名的选举中，我们考虑每对候选人之间的正面交锋结果，并形成一个标记的有向图，称为边际图，其中包含每个候选人对每个候选人的胜利幅度。其他候选人。开发投票方法的一个核心问题是该图中可能存在循环，其中候选人 $\mathsf{A}$ 击败候选人 $\mathsf{B}$，$\mathsf{B}$ 击败 $\mathsf{C} $，而 $\mathsf{C}$ 打败了 $\mathsf{A}$。在本文中，我们应用中心极限定理、图同源性和线性代数来分析大量选民发生这种情况的可能性。有大量关于分析多数获胜者概率的文献；我们的分析更细粒度。