An Analysis of Random Elections with Large Numbers of Voters

09/07/2020 โˆ™ by Matthew Harrison-Trainor, et al. โˆ™ 0 โˆ™

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 ๐–  defeats candidate ๐–ก, ๐–ก defeats ๐–ข, and ๐–ข defeats ๐– . 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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