Condorcet Paradox

Structure of the paradox

Suppose voter 1 ranks A>B>C, voter 2 ranks B>C>A and voter 3 ranks C>A>B. Comparing A and B, voters 1 and 3 choose A, so A wins; between B and C, voters 1 and 2 choose B; and between C and A, voters 2 and 3 choose C. The collective preference therefore cycles, A>B>C>A, and there is no Condorcet winner that beats every alternative. Condorcet (1785) described the phenomenon in a treatise on the probability of majority decisions, in which he also presented the jury theorem: if each voter is more likely than not to be right, the accuracy of majority rule rises with group size.

Conditions that avoid cycles and generalizations

Black (1948, 1958) showed that if all voters have single-peaked preferences along one dimension, no cycle occurs and the median voter's favorite is the Condorcet winner, the median voter theorem. Sen (1966) generalized this into a value-restriction condition. Arrow (1951) extended the paradox into a far more general result, proving in his impossibility theorem that no collective decision rule other than dictatorship satisfies a small set of reasonable conditions at once. Gehrlein (1983) calculated the probability of cycles as a function of the numbers of voters and alternatives.

Agenda control and debates about democracy

Plott (1967) and McKelvey (1976) showed that in policy spaces of two or more dimensions majority rule equilibria almost never exist, and that whoever sets the agenda can order votes to reach a desired outcome. Shepsle (1979) explained that institutions restricting the agenda, such as committee systems, create structure-induced equilibria. Riker (1982) argued on this basis that the very notion of a popular will is unstable, but Mackie (2003) countered that cycles are rare in actual cases. Regenwetter et al. (2006) reported that cycles are almost never observed in large election data, while Kurrild-Klitgaard (2001) documented a rare cycle in Danish opinion poll data.

FAQ

What is the Condorcet paradox in simple terms?

Even if three people each have a clear ranking, comparing options two at a time by majority vote can produce a rock–paper–scissors result in which no option comes out on top.

Does it happen often in real elections?

In theory the chance rises with more voters and alternatives, but in large real elections preferences often share a common structure and cycles are rarely observed.

Sources reviewed: 2026-10-26

References

1. Condorcet, M. J. A. N. de C., marquis de. (1785). Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix. Paris: Imprimerie Royale.
2. Black, D. (1948). On the rationale of group decision-making. Journal of Political Economy, 56(1), 23–34.
3. Black, D. (1958). The Theory of Committees and Elections. Cambridge: Cambridge University Press.
4. Arrow, K. J. (1951). Social Choice and Individual Values. New York: Wiley.
5. Sen, A. K. (1966). A possibility theorem on majority decisions. Econometrica, 34(2), 491–499.
6. McKelvey, R. D. (1976). Intransitivities in multidimensional voting models and some implications for agenda control. Journal of Economic Theory, 12(3), 472–482.
7. Plott, C. R. (1967). A notion of equilibrium and its possibility under majority rule. American Economic Review, 57(4), 787–806.
8. Riker, W. H. (1982). Liberalism Against Populism: A Confrontation Between the Theory of Democracy and the Theory of Social Choice. San Francisco: W. H. Freeman.
9. Shepsle, K. A. (1979). Institutional arrangements and equilibrium in multidimensional voting models. American Journal of Political Science, 23(1), 27–59.
10. Gehrlein, W. V. (1983). Condorcet's paradox. Theory and Decision, 15(2), 161–197.
11. Mackie, G. (2003). Democracy Defended. Cambridge: Cambridge University Press.
12. Regenwetter, M., Grofman, B., Marley, A. A. J., & Tsetlin, I. M. (2006). Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications. Cambridge: Cambridge University Press.
13. Kurrild-Klitgaard, P. (2001). An empirical example of the Condorcet paradox of voting in a large electorate. Public Choice, 107(1–2), 135–145.