¹ Obviously, data regarding first place votes in a three-candidate election will be biased due to strategic voting. This incidence will have a tendency to understate the likelihood of a cycle.

² de Caritat, M. J. A. N., de Condorcet, Marquis, Essai sur l’application de l’analyse à la probabilité des decisions rendues à la pluralité des voix (Paris: Imprimerie Royale, 1785)Google Scholar; Black, Duncan, ‘On the Rationale of Group Decision Making’, Journal of Political Economy, 56 (1948), 23–34CrossRefGoogle Scholar; Arrow, Kenneth J., Social Choice and Individual Values, 2nd edn (New Haven, Conn.: Yale University Press, 1963)Google Scholar; Sen, Amartya, Collective Choice and Social Welfare (San Francisco: Holden-Day, 1970)Google Scholar; Fishburn, Peter C., ‘Condorcet Social Choice Functions’, SIAM Journal of Applied Mathematics, 33 (1977), 469–489CrossRefGoogle Scholar.

³ McKelvey, Richard D., ‘Intransitivities in Multidimensional Voting Models and Some Implications for Agenda Control’, Journal of Economic Theory, 12 (1976), 472–482CrossRefGoogle Scholar; Schofield, Norman, ‘Instability of Simple Dynamic Games’, Review of Economics Studies, 45 (1978), 575–594CrossRefGoogle Scholar.

⁴ Jones, Bradford, Radcliff, Benjamin, Taber, Charles S. and Timpone, Richard, ‘Condorcet Winners and the Paradox of Voting: Probability Calculations for Weak Preference Orders’, American Political Science Review, 89 (1995), 137–144CrossRefGoogle Scholar.

⁵ Tullock, Gordon, ‘The General Irrelevance of the General Impossibility Theorem’, Quarterly Journal of Economics, 81 (1967), 256–270CrossRefGoogle Scholar, p. 256; Niemi, Richard G., ‘Majority Decision-Making with Partial Unidimensionality’, American Political Science Review, 63 (1969), 488–497CrossRefGoogle Scholar; Niemi, Richard, ‘Why So Much Stability? Another Opinion’, Public Choice, 42 (1983), 261–270Google Scholar; Richards, Whitman, McKay, Brendan D. and Richards, Diana, ‘The Probability of Collective Choice with Shared Knowledge Structures’, Journal of Mathematical Psychology 46 (2002), 338–351CrossRefGoogle Scholar.

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⁹ Regenwetter, Michel, Grofman, Bernard, Marley, A. A. J. and Tsetlin, Ilia M., Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications (Cambridge: Cambridge University Press, 2006)Google Scholar.

¹⁰ Regenwetter et al., Behavioral Social Choice.

¹¹ Black, ‘On the Rationale of Group Decision Making’. For an expansive treatment, see Black, Duncan, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958)Google Scholar.

¹² Arrow, Social Choice and Individual Values.

¹³ Black, , The Theory of Committees and Elections, pp. 49–51Google Scholar and 173–4; Riker, William H., ‘Voting and the Summation of Preferences: An Interpretive Bibliographical Review of Selected Developments during the Last Decade’, American Political Science Review, 55 (1961), 900–911Google Scholar; Campbell, Colin D. and Tullock, Gordon, ‘A Measure of the Importance of Cyclical Majorities’, Economic Journal, 75 (1965), 853–856CrossRefGoogle Scholar; Klahr, David, ‘A Computer Simulation of the Paradox of Voting’, American Political Science Review, 60 (1966), 384–390CrossRefGoogle Scholar; Niemi, Richard and Weisberg, Herbert, ‘A Mathematical Solution for the Probability of the Paradox of Voting’, Behavioral Science, 13 (1968), 317–323CrossRefGoogle Scholar; Niemi, ‘Majority Decision-Making with Partial Unidimensionality’; DeMeyer, Frank and Plott, Charles R., ‘The Probability of a Cyclical Majority’, Econometrica, 38 (1970), 345–354CrossRefGoogle Scholar.

¹⁴ Black, The Theory of Committees and Elections.

¹⁵ Jones et al., ‘Condorcet Winners and the Paradox of Voting’; Richards et al., ‘The Probability of Collective Choice with Shared Knowledge Structures’.

¹⁶ Black, , The Theory of Committees and Elections, pp. 49–51Google Scholar.

¹⁷ DeMeyer and Plott, ‘The Probability of a Cyclical Majority’.

¹⁸ Early statements to this effect come from Campbell and Tullock, ‘A Measure of the Importance of Cyclical Majorities’; Niemi, ‘Majority Decision-Making with Partial Unidimensionality’.

¹⁹ Tullock, Gordon, ‘The General Irrelevance of the General Impossibility Theorem’, Quarterly Journal of Economics, 81 (1967), 256–270CrossRefGoogle Scholar; Tullock, Gordon, ‘Why So Much Stability’, Public Choice, 37 (1981), 189–205CrossRefGoogle Scholar; Tullock, Gordon, ‘Why No Cycles?’ American Economic Journal, 28 (2000), 1–12Google Scholar; Niemi, ‘Majority Decision-Making with Partial Unidimensionality’; Feld, Scott L. and Grofman, Bernard, ‘Who’s Afraid of the Big Bad Cycle?’ Journal of Theoretical Politics, 4 (1992), 231–237CrossRefGoogle Scholar, p. 231; Grofman, ‘Public Choice, Civic Republicanism, and American Politics’; Richards et al., ‘The Probability of Collective Choice with Shared Knowledge Structures’; Adams, James and Adams, Ernest W., The Geometry of Elections (Palo Alto, Calif.: CSLI Publications, 2006)Google Scholar; Regenwetter et al., Behavioral Social Choice.

²⁰ Saari, Donald, Geometry of Voting (Berlin: Springer-Verlag, 1994)CrossRefGoogle Scholar.

²¹ For clarification, in the forward cycle, the points of the region on the graph are shown in Table 1. A forward cycle exists when x < [N(A) + N(B) − N(C)]/2. The length of that region is found by subtracting [N(A) + N(B) − N(C)]/2 from N(A). The height is where y > [N(B) − N(A) + N(C)]/2 and the width is where r > [N(B) + N(A) − N(C)]/2. The volume is found by multiplying these together.

²² ((N(B) − N(A) + N(C))/2)(N(A) − (N(A) + N(B) − N(C))/2))(N(B) + N(A) − N(C))/2) + (N(C) − (N(B) − N(A) + N(C))/2)((N(A) + N(B) − N(C))/2)(N(B) − (N(B) + N(A) − N(C))/2) = 1/8 (N(C) − N(B) + N(A))(N(A) + N(B) − N(C)) (N(B) − N(A) + N(C)) + 1/8 (N(C) − N(B) + N(A))(N(A) + N(A) − N(C)) (N(B) − N(A) + N(C)) = 1/8 (2((N(C) − N(B) + N(A))(N(A) + N(B) − N(C))(N(B) − N(A) + N(C)) = 1/4(N(C) − N(B) + N(A))(N(A) + N(B) − N(C))(N(B) − N(A) + N(C)).

²³ This is a familiar result shown in, for example, Niemi, ‘Majority Decision-Making with Partial Unidimensionality’.

²⁴ Average vote percentages are Labour, 40.9 per cent, Liberal/Alliance/Liberal Democrat, 12.2 per cent and Conservative 41.1 per cent.

²⁵ As observable in Budge, Ian, Klingemann, Hans-Dieter, Volkens, Andrea, Bara, Judith and Tanenbaum, Eric, Mapping Policy Preferences: Estimates for Parties, Governments and Electors 1945–1998 (Oxford: Oxford University Press, 2001), p. 25Google Scholar.

²⁶ These data were originally collected by the 2004 European Election Study research group, made possible by a number of grants. The principal investigators were Herman Schmidt and Matthew Loveless in collaboration with others, including James Tilley. The data for Britain are for Great Britain only, not including Northern Ireland. Neither the original collectors of the data nor their sponsors bear responsibility for the analyses or interpretations reported here. The data are available on the European Election Study homepage (http://www.europeanelectionstudies.net/) and from the Central Archive for Empirical Social Research (ZA) at the University of Cologne, Germany.

²⁷ The questionnaire asked for a retrospective vote report and followed with this question (quoted here from the English Master Questionnaire, Part IV of the ESS 2004 Codebook): ‘And if there was a general election tomorrow, which party would you vote for?’ Each respondent was next prompted and asked this: ‘We have a number of parties in [country] each of which would like to get your vote. How likely is it that you will ever vote for the following parties? Please specify your views on a 10-point scale where 1 means “not at all probable” and 10 means “very probable”. If you think of [Party 1], what mark out of 10 best describes how probable it is that you will ever vote for [Party 1]?’

²⁸ We could also have used the probability statements for respondents refusing to indicate a preference or saying they ‘don’t know’. Their inclusion does not change the collective preference order based on head-to-head party competition reported in Table 2.

²⁹ Regenwetter et al., Behavioral Social Choice.