Advertisement

Public Choice

, Volume 158, Issue 3–4, pp 311–330 | Cite as

On the empirical relevance of Condorcet’s paradox

  • Adrian Van Deemen
Article

Abstract

Condorcet’s paradox occurs when there is no alternative that beats every other alternative by majority. The paradox may pose real problems to democratic decision making such as decision deadlocks and democratic paralysis. However, its relevance has been discussed again and again since the celebrated works of Arrow (Social choice and individual values, 1963) and Black (The theory of committees and elections, 1958). The discussion varies from one extreme to the other: from very relevant to practically irrelevant. This paper tries to bring more clarity to the discussion by reviewing the literature on the empirical relevance of Condorcet’s paradox. Since a definition of the paradox for even numbers of voters and alternatives, and for weak voter preferences is missing in the literature, we first define the paradox clearly and simply. Then, three topics are investigated, namely domain conditions, culture and the likelihood of the paradox, and the empirical detection of the paradox. Domain conditions express regularities in voter-preference profiles that prevent the paradox. Frequent observations of these domain conditions would make Condorcet’s paradox empirically less important. Cultures define probability distributions over the set of voter preferences. Observation of cultures might be a first step to indicate the relevance of the paradox. The empirical detection of the paradox speaks for itself; we will try to identify the number of observations of the paradox so far. The overall conclusion is that the empirical relevance of Condorcet’s paradox is still unsettled.

Keywords

Condorcet’s paradox Majority rule Majority games Social choice Voting 

Notes

Acknowledgements

The author would like to thank the five anonymous referees for their critical remarks, useful comments, and valuable suggestions for improvements. Of course, the author is responsible for the remaining errors and omissions.

References

  1. Abramson, P., Aldrich, J., Paolini, P., & Rohde, D. (1995). Third-party and independent candidates in American politics. Political Science Quarterly, 110, 349–367. CrossRefGoogle Scholar
  2. Arrow, K. (1963). Social choice and individual values (2nd ed.). New Haven: Yale University Press. Google Scholar
  3. Beck, J. (1997). Voting cycles in business curriculum reform: a note. The American Economic Review, 41, 83–88. Google Scholar
  4. Bianco, W., Lynch, M., Miller, G., & Sened, I. (2006). “A theory waiting to be discovered and used”: a reanalysis of canonical experiments on majority decision making. The Journal of Politics, 68, 838–851. CrossRefGoogle Scholar
  5. Bjurulf, B. H., & Niemi, R. G. (1978). Strategic voting in Scandinavian parliaments. Scandinavian Political Studies, 1, 5–22. CrossRefGoogle Scholar
  6. Black, D. (1958). The theory of committees and elections. Cambridge: Cambridge University Press. Google Scholar
  7. Blydenburgh, J. C. (1971). The closed rule and the paradox of voting. The Journal of Politics, 33, 57–71. CrossRefGoogle Scholar
  8. Bochsler, D. (2010). The Marquis de Condorcet goes to Bern. Public Choice, 144, 119–131. CrossRefGoogle Scholar
  9. Browne, E. C., & Hamm, K. E. (1996). Legislative politics and the paradox of voting: electoral reform in the Fourth Republic France. British Journal of Political Science, 26, 165–198. CrossRefGoogle Scholar
  10. Campbell, C. D., & Tullock, G. (1965). A measure of importance of cyclical majorities. Economic Journal, 75, 853–857. CrossRefGoogle Scholar
  11. Chamberlin, J. R., Cohen, J. L., & Coombs, C. H. (1984). Social choice observed: five presidential elections of the American psychological association. The Journal of Politics, 46, 479–502. CrossRefGoogle Scholar
  12. Condorcet, M. (1785). Essai sur l’application de l’analyse a la probabilité des decisions rendues a la pluralité des voix. In M. Condorcet (Ed.), Sur les élections et autres textes. Corpus des Oeuvres de philosophie en langue Française. Paris: Librairie Arthème Fayard. (1986). Google Scholar
  13. Dahl, R. (1956). A preface to democratic theory. Chicago: University of Chicago Press. Google Scholar
  14. Dahl, R. (1989). Democracy and its critics. New Haven: Yale University Press. Google Scholar
  15. Davis, O. M., DeGroot, H., & Hinich, M. J. (1972). Social preference orderings and majority rule. Econometrica, 40, 147–157. CrossRefGoogle Scholar
  16. De Meyer, F., & Plott, C. (1970). The probability of a cyclical majority. Econometrica, 38, 345–354. CrossRefGoogle Scholar
  17. Dietz, H. A., & Goodman, M. J. (1987). An empirical analysis of preferences in the 1983 multicandidate Peruvian mayoral election. American Journal of Political Science, 31, 281–295. CrossRefGoogle Scholar
  18. Dobra, J. (1983). An approach to empirical studies in voter paradoxes: an update and extension. Public Choice, 41, 241–250. CrossRefGoogle Scholar
  19. Dobra, J., & Tullock, G. (1981). An approach to the empirical measures of voting paradoxes. Public Choice, 41, 193–194. Google Scholar
  20. Dodgson, C. L. (1876). A method of taking votes on more than two issues. In D. Black (Ed.) (1958). Google Scholar
  21. Dyer, J. S., & Miles, R. F. (1976). An application of collective choice theory to the selection of trajectories for the Mariner Jupiter/Saturn project. Operational Research, 24, 220–244. CrossRefGoogle Scholar
  22. Enelow, J., & Hinich, M. (1984). The spatial theory of voting. Cambridge: Cambridge University Press. Google Scholar
  23. Feld, S., & Grofman, B. (1987). Necessary and sufficient conditions for a majority winner in n-dimensional spatial voting games: an intuitive geometric approach. American Journal of Political Science, 31, 709–728. CrossRefGoogle Scholar
  24. Feld, S. L., & Grofman, B. (1992). Who’s afraid of the big bad cycle? Evidence from 36 elections. Journal of Theoretical Politics, 4, 231–237. CrossRefGoogle Scholar
  25. Fiorina, M., & Plott, C. (1978). Committee decisions under majority rule: an experimental study. American Political Science Review, 72, 575–598. CrossRefGoogle Scholar
  26. Fishburn, P. (1973a). The theory of social choice. Princeton: Princeton University Press. Google Scholar
  27. Fishburn, P. (1973b). Voter concordance, simple majority, and group decision methods. Behavioral Science, 18, 364–376. CrossRefGoogle Scholar
  28. Fishburn, P., & Little, J. D. C. (1988). An experiment in approval voting. Management Sciences, 34, 555–568. CrossRefGoogle Scholar
  29. Flanagan, T. (1997). The staying power of the legislative status quo: collective choice in Canada’s parliament after Morgenthaler. Canadian Journal of Political Science, 30, 31–53. CrossRefGoogle Scholar
  30. Flood, M. (1955). A group preference experiment. In Mathematical models of human behavior (pp. 130–148). Stanford: Dunlap and Associates. Google Scholar
  31. Gaertner, W. (2001). Domain conditions in social choice theory. Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  32. Garman, M. B., & Kamien, M. I. (1968). The paradox of voting: probability calculations. Behavioral Science, 13, 306–316. CrossRefGoogle Scholar
  33. Gaubatz, K. T. (1995). Intervention and intransitivity: public opinion, social choice, and the use of military forces abroad. World Politics, 47, 534–554. CrossRefGoogle Scholar
  34. Gehrlein, W. V. (1983). Condorcet’s paradox. Theory and Decision, 15, 161–197. CrossRefGoogle Scholar
  35. Gehrlein, W. V. (2004). The likelihood of complete breakdown by majority rule. In: Proceedings of National Decision Sciences Institute, Boston, MA (pp. 411–416). Google Scholar
  36. Gehrlein, W. V. (2006). Condorcet’s paradox. Berlin: Springer. Google Scholar
  37. Gehrlein, W. V., & Fishburn, P. (1976). The probability of the paradox of voting: a computable solution. Journal of Economic Theory, 13, 14–25. CrossRefGoogle Scholar
  38. Gehrlein, W. V., & Lepelley, D. (2011). Voting paradoxes and group coherence. Berlin: Springer. CrossRefGoogle Scholar
  39. Hsieh, J. F., Niou, E. M., & Paolino, P. (1997). Analytical representations of probabilities under the IAC condition. Social Choice and Welfare, 17, 143–156. Google Scholar
  40. Inada, K. (1964). A note on the simple majority decision rule. Econometrica, 32, 316–338. CrossRefGoogle Scholar
  41. Inada, K. (1969). On the simple majority decision rule. Econometrica, 37, 490–506. CrossRefGoogle Scholar
  42. Jones, B., Radcliff, B., Taber, C., & Timpone, T. (1995). Condorcet winners and the paradox of voting: probability calculations for weak preference orders. American Political Science Review, 89, 137–147. CrossRefGoogle Scholar
  43. Kemeny, J., & Snell, J. (1962). Mathematical models in the social sciences. Cambridge: MIT Press. Google Scholar
  44. Klahr, D. (1966). A computer simulation of the paradox of voting. The American Political Science Review, 60, 384–390. CrossRefGoogle Scholar
  45. Kurrild-Klitgaard, P. (2001). An empirical example of the Condorcet paradox of voting in a large electorate. Public Choice, 107, 135–145. CrossRefGoogle Scholar
  46. Kurrild-Klitgaard, P. (2005). Individ, Stat og Marked: Studier i Rationalitet og Politik, København: Politiske Studier. Available at SSRN. http://ssrn.com/abstract=1972300 or doi: 10.2139/ssrn.1972300.
  47. Kurrild-Klitgaard, P. (2008). Voting paradoxes under proportional representation: evidence from eight Danish elections. Scandinavian Political Studies, 31, 242–267. CrossRefGoogle Scholar
  48. Lagerspetz, E. (1997). Social choice in the real world II: cyclical preferences and strategic voting in the Finnish presidential elections. Scandinavian Political Studies, 20, 53–67. CrossRefGoogle Scholar
  49. Lepelley, D., & Martin, M. (2001). Condorcet’s paradox for weak preference orderings. European Journal of Political Economy, 17, 163–177. CrossRefGoogle Scholar
  50. Mackie, G. (2003). Democracy defended. Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  51. May, R. M. (1971). Some mathematical remarks on the paradox of voting. Behavioral Science, 16, 143–151. CrossRefGoogle Scholar
  52. McKelvey, R. (1976). Intransitivities in multidimensional voting models and some implications for agenda control. Journal of Economic Theory, 12, 472–482. CrossRefGoogle Scholar
  53. McKelvey, R. (1979). General conditions for global intransitivities in formal voting models. Econometrica, 47, 1085–1112. CrossRefGoogle Scholar
  54. McKelvey, R., & Ordeshook, P. (1990). A decade of experimental research on spatial models. In J. Enelow & M. Hinich (Eds.), Advances in the spatial theory of voting (pp. 99–145). Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  55. Miller, N. R. (2007). In search of the uncovered set. Political Analysis, 15, 21–45. CrossRefGoogle Scholar
  56. Morse, J. R. (1997). Constitutional rules, political accidents, and the course of history: new light on the annexation of Texas. Independent Review, 2, 173–201. Google Scholar
  57. Nakamura, K. (1975). The core of a simple game with ordinal preferences. International Journal of Game Theory, 4, 95–104. CrossRefGoogle Scholar
  58. Nakamura, K. (1979). The vetoers in a simple game with ordinal preferences. International Journal of Game Theory, 8, 55–61. CrossRefGoogle Scholar
  59. Neufeld, J. L., Hausman, W. J., & Rapoport, R. B. (1994). A paradox of voting. Cyclical majorities and the case of muscle shoals. Political Research Quarterly, 47, 423–438. Google Scholar
  60. Niemi, R. G. (1969). Majority decision-making with partial unidimensionality. The American Political Science Review, 63, 488–497. CrossRefGoogle Scholar
  61. Niemi, R. G. (1970). The occurrence of the paradox of voting in university elections. Public Choice, 8, 91–100. CrossRefGoogle Scholar
  62. Niemi, R. G., & Weisberg, H. F. (1968). A mathematical solution for the probability of the paradox of voting. Behavioral Science, 13, 317–323. CrossRefGoogle Scholar
  63. Norpoth, H. (1979). The parties come to order! Dimensions of preferential choice in the West German electorate. The American Political Science Review, 73, 724–736. CrossRefGoogle Scholar
  64. Nurmi, H. (1999). Voting paradoxes and how to deal with them. Berlin: Springer. CrossRefGoogle Scholar
  65. Owen, G. (1990). Stable outcomes in spatial voting games. Mathematical Social Sciences, 19, 269–279. CrossRefGoogle Scholar
  66. Owen, G., & Shapley, L. S. (1989). Optimal location of candidates in ideological space. International Journal of Game Theory, 18, 339–356. CrossRefGoogle Scholar
  67. Pattanaik, P. (1971). Voting and collective choice. Cambridge: Cambridge University Press. Google Scholar
  68. Peleg, B. (1984). Game theoretic analysis of voting in committees. Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  69. Plott, C. (1967). A notion of equilibrium and its possibility under majority rule. The American Economic Review, 57, 787–806. Google Scholar
  70. Radcliff, B. (1994). Collective preferences in presidential elections. Electoral Studies, 13, 50–57. CrossRefGoogle Scholar
  71. Regenwetter, M., & Grofman, B. (1998). Approval voting, Borda winners, and Condorcet winners: evidence from seven elections. Management Science, 44, 520–533. CrossRefGoogle Scholar
  72. Regenwetter, M., Adams, J., & Grofman, B. (2002a). On the (sample) Condorcet efficiency of majority rule: an alternit e view of majority cycles and social homogeneity. Theory and Decision, 53, 153–186. CrossRefGoogle Scholar
  73. Regenwetter, M., Grofman, B., & Marley, A. A. J. (2002b). On the model dependence of majority preference relations reconstructed from ballot or survey data. Mathematical Social Sciences, 43, 451–466. CrossRefGoogle Scholar
  74. Regenwetter, M. B., Marly, A. A. J., & Grofman, B. (2003). General concepts of value restriction and preference majority. Social Choice and Welfare, 21, 149–173. CrossRefGoogle Scholar
  75. Regenwetter, M., Grofman, B., Marley, A., & Tsetlin, I. (2006). Behavioral social choice. Cambridge: Cambridge University Press. Google Scholar
  76. Regenwetter, M., Kim, A., Kantor, A., & Ho, M. (2007). The unexpected empirical consensus among consensus methods. Psychological Science, 18, 629–635. CrossRefGoogle Scholar
  77. Riker, W. H. (1958). The paradox of voting and congressional rules for voting on amendments. The American Political Science Review, 52, 349–366. CrossRefGoogle Scholar
  78. Riker, W. H. (1965). Arrow’s theorem and some examples of the paradox of voting. In J. M. Claunch (Ed.), Mathematical applications in political science, Dallas (pp. 41–60). Google Scholar
  79. Riker, W. H. (1980). Implications from the disequilibrium of majority rule for the study of institutions. American Political Science Review, 74(2), 432–446. CrossRefGoogle Scholar
  80. Riker, W. H. (1982). Liberalism against populism. San Francisco: Freeman. Google Scholar
  81. Rosen, M. D., & Sexton, R. J. (1993). Irrigation districts and water markets: an application of cooperative decision making. Land Economics, 69, 39–54. CrossRefGoogle Scholar
  82. Sen, A. (1966). A possibility theorem on majority decisions. Econometrica, 34, 491–496. CrossRefGoogle Scholar
  83. Sen, A. (1970). Collective choice and social welfare. San Fransisco: Holden Day. Google Scholar
  84. Sen, A., & Pattanaik, P. (1969). Necessary and sufficient conditions for rational choice under majority decision. Journal of Economic Theory, 1, 178–202. CrossRefGoogle Scholar
  85. Shapley, L. S. (1963). Simple games: an outline of the descriptive theory. Behavioral Science, 7, 59–66. CrossRefGoogle Scholar
  86. Smith, W. D. (2009). The Romanian 2009 presidential election featured Condorcet cycles. http://www.rangevoting.org/Romania2009.html.
  87. Stensholt, E. (1999). Voteringens kvaler: flypkasen I Stortinget 8 Okotber 1992. Sosialøkonomen, 4, 28–40. Google Scholar
  88. Taplin, R. H. (1997). The statistical analysis of preference data. Applied Statistics, 46, 493–512. Google Scholar
  89. Taylor, M. (1968). Graph-theoretic approaches to the theory of social choice. Public Choice, 4, 35–47. CrossRefGoogle Scholar
  90. Taylor, A. (1997). A glimpse of impossibility: Kenneth Arrow’s impossibility theory and voting. Perspectives on Political Science, 26, 23–26. CrossRefGoogle Scholar
  91. Tideman, N. (2006). Collective decisions and voting. Chippenham: Ashgate. Google Scholar
  92. Tideman, N. (2012). Personal communication. March 23, 2012 (by email). Google Scholar
  93. Toda, M., Sugiyama, K., & Tagawa, S. (1982). A method for aggregating ordinal assessments by a majority decision rule. Mathematical Social Sciences, 3, 227–242. CrossRefGoogle Scholar
  94. Truchon, M. (1998). Rating skating and the theory of social choice. Université Laval. Unpublished. Google Scholar
  95. Van Dam, M. J. (1998). The purple paradox: decision making on the MUB. Acta Politica, 33, 77–85. Google Scholar
  96. Van Deemen, A. (1997). Coalition formation and social choice. Dordrecht: Kluwer Academic. CrossRefGoogle Scholar
  97. Van Deemen, A. (1999). The probability of the paradox of voting for weak preference orderings. Social Choice and Welfare, 10, 171–182. CrossRefGoogle Scholar
  98. Van Deemen, A., & Saiz, H. (2010). Extremal restriction, Condorcet sets, and majority decision making. In A. Van Deemen & A. Rusinowska (Eds.), Collective decision making (pp. 69–85). CrossRefGoogle Scholar
  99. Van Deemen, A., & Vergunst, N. (1998). Empirical evidence of paradoxes of voting in Dutch elections. Public Choice, 97, 475–490. CrossRefGoogle Scholar
  100. Vergunst, N. (1996). Besluitvorming over kerncentrale Borssele: een analyse van de stemparadox in de Nederlandse politiek. Acta Politica, 31, 209–228. Google Scholar
  101. Wilson, S. (2003). O boy. Sun Journal, July 3. Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Management ResearchRadboud University NijmegenNijmegenThe Netherlands

Personalised recommendations