An analysis of random elections with large numbers of voters

Matthew Harrison-Trainor

Mathematical Social Sciences, 2022, vol. 116, issue C, 68-84

Abstract: 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 A defeats candidate B, B defeats C, and C defeats A. It is known that such cycles are unlikely to occur. Under the Impartial Culture assumption, in a random election with three candidates and a very large number of voters there is a 91.23% chance of avoiding a cycle. By studying the geometry of the space of random elections, we give a mathematical explanation of why this is the case. Our main result is that margin graphs that are more cyclic in a certain precise sense are less likely to occur. This connects the probabilistic study of voting methods to Zwicker’s analysis of Condorcet’s paradox in terms of cycles and cuts.

Keywords: Condorcet paradox; Random election; Multivariate normal distribution (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:matsoc:v:116:y:2022:i:c:p:68-84

DOI: 10.1016/j.mathsocsci.2022.01.002

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