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The Probability of Conflicts in a US Presidential Type Election


  • Dominique Lepelley

    () (CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR1 - Université de Rennes 1 - UNIV-RENNES - Université de Rennes - CNRS - Centre National de la Recherche Scientifique)

  • Vincent Merlin

    () (CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR1 - Université de Rennes 1 - UNIV-RENNES - Université de Rennes - CNRS - Centre National de la Recherche Scientifique)

  • M. Feix
  • J.-L. Rouet


In a two candidate election, it might be that a candidate wins in a majority of districts while he gets less vote than his opponent in the whole country. In Social Choice Theory, this situation is known as the compound majority paradox, or the referendum paradox. Although occurrences of such paradoxical results have been observed worldwide in political elections (e.g. United States, United Kingdom, France), no study evaluates theoretically the likelihood of such situations. In this paper, we propose four probability models in order to tackle this issue, for the case where each district has the same population. For a divided electorate, our results prove that the likelihood of this paradox rapidly tends to 20% when the number of districts increases. This probability decreases with the number of states when a candidate receives significatively more vote than his opponent over the whole country. Copyright Springer-Verlag Berlin/Heidelberg 2004
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  • Dominique Lepelley & Vincent Merlin & M. Feix & J.-L. Rouet, 2004. "The Probability of Conflicts in a US Presidential Type Election," Post-Print halshs-00070893, HAL.
  • Handle: RePEc:hal:journl:halshs-00070893
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    Cited by:

    1. Marek M. Kaminski, 2015. "Empirical examples of voting paradoxes," Chapters, in: Jac C. Heckelman & Nicholas R. Miller (ed.), Handbook of Social Choice and Voting, chapter 20, pages 367-387, Edward Elgar Publishing.
    2. Laurent, Thibault & Le Breton, Michel & Lepelley, Dominique & de Mouzon, Olivier, 2017. "Exploring the Effects on the Electoral College of National and Regional Popular Vote Interstate Compact: An Electoral Engineering Perspective," TSE Working Papers 17-861, Toulouse School of Economics (TSE), revised May 2018.
    3. Mostapha Diss & Vincent Merlin, 2010. "On the stability of a triplet of scoring rules," Theory and Decision, Springer, vol. 69(2), pages 289-316, August.
    4. Le Breton, Michel & Lepelley, Dominique & Macé, Antonin & Merlin, Vincent, 2017. "Le mécanisme optimal de vote au sein du conseil des représentants d’un système fédéral," L'Actualité Economique, Société Canadienne de Science Economique, vol. 93(1-2), pages 203-248, Mars-Juin.
    5. Le Breton, Michel & Van Der Straeten, Karine, 2014. "Influence Vs. Utility in the Evaluation of Voting Rules: A New Look at the Penrose Formula," TSE Working Papers 14-511, Toulouse School of Economics (TSE).
    6. Nicolas Houy, 2006. "La Constitution européenne est 50,13 %-stable. Une note comparative sur la stabilité des Constitutions," Revue économique, Presses de Sciences-Po, vol. 57(1), pages 123-134.
    7. Kurrild-Klitgaard, Peter, 2018. "Trump, Condorcet and Borda: Voting paradoxes in the 2016 Republican presidential primaries," European Journal of Political Economy, Elsevier, vol. 55(C), pages 29-35.
    8. Laurent, Thibault & Le Breton, Michel & Lepelley, Dominique & de Mouzon, Olivier, 2016. "The Theoretical Shapley-Shubik Probability of an Election Inversion in a Toy Symmetric Version of the U.S. Presidential Electoral System," TSE Working Papers 16-671, Toulouse School of Economics (TSE), revised Feb 2018.
    9. Olivier Mouzon & Thibault Laurent & Michel Breton & Dominique Lepelley, 2019. "Exploring the effects of national and regional popular vote Interstate compact on a toy symmetric version of the Electoral College: an electoral engineering perspective," Public Choice, Springer, vol. 179(1), pages 51-95, April.
    10. Kazuya Kikuchi & Yukio Koriyama, 2019. "The Winner-Take-All Dilemma," ISER Discussion Paper 1059r, Institute of Social and Economic Research, Osaka University, revised Dec 2019.
    11. Serguei Kaniovski & Alexander Zaigraev, 2018. "The probability of majority inversion in a two-stage voting system with three states," Theory and Decision, Springer, vol. 84(4), pages 525-546, June.
    12. Merlin, Vincent & Valognes, Fabrice, 2004. "The impact of indifferent voters on the likelihood of some voting paradoxes," Mathematical Social Sciences, Elsevier, vol. 48(3), pages 343-361, November.
    13. Lepelley, Dominique & Merlin, Vincent & Rouet, Jean-Louis, 2011. "Three ways to compute accurately the probability of the referendum paradox," Mathematical Social Sciences, Elsevier, vol. 62(1), pages 28-33, July.
    14. Nicholas Miller, 2012. "Why the Electoral College is good for political science (and public choice)," Public Choice, Springer, vol. 150(1), pages 1-25, January.
    15. Rafael Treibich & Martin Van der linden, 2017. "Trump trumps Bush," Vanderbilt University Department of Economics Working Papers 17-00014, Vanderbilt University Department of Economics.
    16. Wilson, Mark C. & Pritchard, Geoffrey, 2007. "Probability calculations under the IAC hypothesis," Mathematical Social Sciences, Elsevier, vol. 54(3), pages 244-256, December.
    17. Sebastian Bervoets & Vincent Merlin, 2012. "Gerrymander-proof representative democracies," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(3), pages 473-488, August.
    18. Nicholas Miller, 2014. "The Alternative Vote and Coombs Rule versus First-Past-the-Post: a social choice analysis of simulated data based on English elections, 1992–2010," Public Choice, Springer, vol. 158(3), pages 399-425, March.
    19. Abraham Diskin & Moshe Koppel, 2010. "Voting power: an information theory approach," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 34(1), pages 105-119, January.
    20. Jac C. Heckelman & Nicholas R. Miller (ed.), 2015. "Handbook of Social Choice and Voting," Books, Edward Elgar Publishing, number 15584.
    21. Kurrild-Klitgaard, Peter, 2011. "Election inversions, coalitions and proportional representation: Examples from Danish elections," MPRA Paper 35302, University Library of Munich, Germany.

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    conflicts; presidential election; United States;


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