Elections and Voting Paradoxes

  • William V. Gehrlein
  • Dominique Lepelley
Part of the Studies in Choice and Welfare book series (WELFARE)


An overview of the importance of the work of both Condorcet and Borda is presented from a historical perspective. Their work is discussed with an emphasis on the possible voting paradoxes that evolved directly from their work: Condorcet’s Paradox, Borda’s Paradox and Condorcet’s Other Paradox. Many other strange outcomes that could happen in elections are shown with examples of other voting paradoxes from later researchers, including: No Show Paradox, Ostrogorski’s Paradox, Majority Paradox and Referendum Paradox. The importance is established for evaluating these paradoxes on the basis of the probability that they might ever actually be observed in practice, to determine if they really pose a significant threat to the stability of elections or if they just reflect some interesting theoretical possibilities with a small number of candidates.


  1. Arrow, K. J. (1963). Social choice and individual values (2nd ed.). New Haven CT: Yale University Press.Google Scholar
  2. Badinter, E., & Badinter, R. (1988). Condorcet: un intellectuel en politique. Paris: Librarie Arthème Fayard.Google Scholar
  3. Baker, K. M. (1975). Condorcet: From natural philosophy to social mathematics. Chicago IL: University of Chicago Press.Google Scholar
  4. Black, D. (1958). The theory of committees and elections. Cambridge: Cambridge University Press.Google Scholar
  5. Brams, S. J., & Fishburn, P. C. (1983). Paradoxes of preferential voting. Mathematics Magazine, 56, 207–214.CrossRefGoogle Scholar
  6. Daunou, P. C. F. (1803/1991). A paper on elections by ballot. In: F. Sommerlad, & I. McLean (Eds.), The political theory of Condorcet II (pp. 235–279). Oxford: University of Oxford Working Paper.Google Scholar
  7. de Borda, J. C. (1784). A paper on elections by ballot. In I. McLean & F. Hewitt (Eds.), Condorcet: Foundations of social choice and political theory (pp. 114–119). Hants: Edward Elgar.Google Scholar
  8. de Condorcet, M. (1785). An essay on the application of probability theory to plurality decision making. In I. McLean & F. Hewitt (Eds.), Condorcet: Foundations of social choice and political theory (pp. 120–130). Hants: Edward Elgar.Google Scholar
  9. de Condorcet, M. (1788a). On the form of decisions made by plurality vote. In I. McLean & F. Hewitt (Eds.), Condorcet: Foundations of social choice and political theory (pp. 157–168). Hants: Edward Elgar.Google Scholar
  10. de Condorcet, M. (1788b). On discovering the plurality will in an election. In I. McLean & F. Hewitt (Eds.), Condorcet: Foundations of social choice and political theory (pp. 148–156). Hants: Edward Elgar.Google Scholar
  11. de Condorcet, M. (1793). A general survey of science - Concerning the application of calculus to the political and moral sciences. In I. McLean & F. Hewitt (Eds.), Condorcet: Foundations of social choice and political theory (pp. 93–98). Hants: Edward Elgar.Google Scholar
  12. de Laplace, P. S. (1795). Analytic theory of probabilities. In: F. Sommerlad, I. McLean (1991, Eds.) The political theory of Condorcet II (pp. 282–286). Oxford: University of Oxford Working Paper.Google Scholar
  13. Dodgson, C. (1885). The principles of parliamentary representation: Postscript to supplement, E. Oxford: Baxter Publisher.Google Scholar
  14. Felsenthal, D. S., & Machover, M. (1992). After two centuries, should Condorcet’s voting procedure be implemented? Behavioral Science, 37, 250–274.CrossRefGoogle Scholar
  15. Fishburn, P. C. (1970). The irrationality of transitivity of social choice. Behavioral Science, 15, 119–123.CrossRefGoogle Scholar
  16. Fishburn, P. C., & Gehrlein, W. V. (1976). Borda’s rule, positional voting, and Condorcet’s simple majority principle. Public Choice, 28, 79–88.CrossRefGoogle Scholar
  17. Gaertner, W. (2005). De jure naturae et gentium: Samuel von Pufendorf’s contribution to social choice theory and economics. Social Choice and Welfare, 25, 231–241.CrossRefGoogle Scholar
  18. Gehrlein, W. V. (1990). Special issue on intransitive preferences. Annals of Operations Research, 23, 235–246.Google Scholar
  19. Gehrlein, W. V. (2006). Condorcet’s paradox. Berlin: Springer.Google Scholar
  20. Granger, G. G. (1956). La mathématique sociale du Marquis de Condorcet. Paris: Presses Universitaires de France.Google Scholar
  21. Huntington, E. V. (1938). A paradox in the scoring of competing teams. Science, 8, 287–288.CrossRefGoogle Scholar
  22. Lagerspetz, E. (1986). Pufendorf on collective decision. Public Choice, 49, 179–182.CrossRefGoogle Scholar
  23. Mascart, J. (2000). La vie et les travaux du chevalier Jean-Charles de Borda (1733–1799), Episodes de la vie scientifique au XVIIIe siècle. Paris: University of Paris-Sorbonne Press.Google Scholar
  24. McLean, I. (1990). The Borda and Condorcet principles: Three medieval applications. Social Choice and Welfare, 7, 99–108.CrossRefGoogle Scholar
  25. McLean, I. (1995). The first golden age of social choice, 1784–1803. In W. Barnett, H. Moulin, M. Salles, & N. Schofield (Eds.), Social choice, welfare, and ethics (pp. 13–36). Cambridge: Cambridge University Press.Google Scholar
  26. McLean, I., & Hewitt, F. (1994). Condorcet: Foundations of social choice and political theory. Hants: Edward Elgar.Google Scholar
  27. Miller, N. (2012). Election inversions by the U.S. Electoral College. In D. S. Felsenthal & M. Machover (Eds.), Electoral systems: Paradoxes, assumptions and procedures (pp. 93–128). Berlin: Springer.CrossRefGoogle Scholar
  28. Neubauer, M. G., Schilling, M., & Zeitlin, J. (2012). Exploring unpopular presidential elections. Working paper, California State University.Google Scholar
  29. Nurmi, H. (1999). Voting paradoxes and how to deal with them. Berlin: Springer.CrossRefGoogle Scholar
  30. Ostrogorski, M. (1902). La démocratie et l’organisation des partis politiques. Paris: Calmann-Levy Publishing.Google Scholar
  31. Riker, W. H. (1958). The paradox of voting and congressional rules for voting on amendments. American Political Science Review, 52, 349–366.CrossRefGoogle Scholar
  32. Riker, W. H. (1961). Voting and the summation of preferences: An interpretive bibliographical review of selected developments during the last decade. American Political Science Review, 55, 900–911.Google Scholar
  33. Rousseau, J. (1762). The social contract, translated and reprinted in 1962 by Penguin Press. England: Harmondsworth.Google Scholar
  34. Saari, D. G. (1995). Basic geometry of voting. Berlin: Springer.CrossRefGoogle Scholar
  35. Todhunter, I. (1865). A history of the mathematical theory of probability. Cambridge: Macmillan.Google Scholar
  36. Wilbour, C. E. (1909). Les Misérables, Vol. 1 (Translated from the French). London: JM Dent and Son.Google Scholar
  37. Young, P. (1995). Optimal voting rules. The Journal of Economic Perspectives, 9, 51–64.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • William V. Gehrlein
    • 1
  • Dominique Lepelley
    • 2
  1. 1.Department of Business AdministrationUniversity of DelawareNewarkUSA
  2. 2.University of La RéunionSaint-Denis, Ile de La RéunionFrance

Personalised recommendations