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An Analysis of Random Elections with Large Numbers of Voters

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  • Matthew Harrison-Trainor

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 $\mathsf{A}$ 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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  • Matthew Harrison-Trainor, 2020. "An Analysis of Random Elections with Large Numbers of Voters," Papers 2009.02979, arXiv.org.
    • Handle: RePEc:arx:papers:2009.02979
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    References listed on IDEAS

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    1. Gehrlein, William V. & Fishburn, Peter C., 1978. "Probabilities of election outcomes for large electorates," Journal of Economic Theory, Elsevier, vol. 19(1), pages 38-49, October.
    2. Riker, William H., 1958. "The Paradox of Voting and Congressional Rules for Voting on Amendments," American Political Science Review, Cambridge University Press, vol. 52(2), pages 349-366, June.
    3. Gehrlein, William V., 1988. "Probability calculations for transitivity of the simple majority rule," Economics Letters, Elsevier, vol. 27(4), pages 311-315.
    4. Kurrild-Klitgaard, Peter, 2001. "An Empirical Example of the Condorcet Paradox of Voting in a Large Electorate," Public Choice, Springer, vol. 107(1-2), pages 135-145, April.
    5. Kramer, Gerald H., 1977. "A dynamical model of political equilibrium," Journal of Economic Theory, Elsevier, vol. 16(2), pages 310-334, December.
    6. De Donder, Philippe & Le Breton, Michel & Truchon, Michel, 2000. "Choosing from a weighted tournament1," Mathematical Social Sciences, Elsevier, vol. 40(1), pages 85-109, July.
    7. Paul B. Simpson, 1969. "On Defining Areas of Voter Choice: Professor Tullock on Stable Voting," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 83(3), pages 478-490.
    8. Markus Schulze, 2011. "A new monotonic, clone-independent, reversal symmetric, and condorcet-consistent single-winner election method," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 36(2), pages 267-303, February.
    9. DeMeyer, Frank & Plott, Charles R, 1970. "The Probability of a Cyclical Majority," Econometrica, Econometric Society, vol. 38(2), pages 345-354, March.
    10. Bhaskar Dutta & Jean-Francois Laslier, 1999. "Comparison functions and choice correspondences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(4), pages 513-532.
    11. Adrian Deemen, 2014. "On the empirical relevance of Condorcet’s paradox," Public Choice, Springer, vol. 158(3), pages 311-330, March.
    12. Peter Craig, 2008. "A new reconstruction of multivariate normal orthant probabilities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 227-243, February.
    13. Gehrlein, William V., 1989. "The probability of intransitivity of pairwise comparisons in individual preference," Mathematical Social Sciences, Elsevier, vol. 17(1), pages 67-75, February.
    14. Truchon, M., 1998. "Figure Skating and the Theory of Social Choice," Papers 9814, Laval - Recherche en Politique Economique.
    15. Gehrlein, William V. & Fishburn, Peter C., 1976. "The probability of the paradox of voting: A computable solution," Journal of Economic Theory, Elsevier, vol. 13(1), pages 14-25, August.
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