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A cyclical majority, as here defined, is a proper subset of the set of intransitive social orderings. Our usage follows Charles L. Dogson's use of the term ‘cyclical majority’ and is equivalent to David Klahr's ‘Type 2 intransitive social ordering’. David Klahr, “A Computer Simulation of the Paradox of Voting,”American Political Science Review 60 (June, 1966) p. 385, and Charles L. Dodgson, “A Method of Taking Votes on More Than Two Issues,” section 2; Dodgson's phamplet is reprinted in Duncan Black,The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1963), pp. 224–234.
Y. Murakami,Logic and Social Choice (New York: Dover Publications Inc., 1968), p. 72. Murakami called this Black's theorem.
William H. Riker, “The Paradox of Voting and Congressional Rules for Voting on Amendments,”American Political Science Review, 52 (June, 1958), pp. 349–66.
Riker, “The Paradox of Voting and Congressional Rules for Voting on Amendments,” pp. 353–4.
Duncan Black, “On the Rationale of Group Decision-Making,”Journal of Political Economy, 56, (February, 1948), pp. 23–34.
Curiously, Murakami and Riker have also erred when counting the number of possible weak preference orderings, although their errors differ in this case. Murakami states that if the number of alternatives is N, then the number of conceivable weak preference orderings is\(\mathop \Sigma \limits_{k = 1}^N \) k!. Murakami,Logic and Social Choice p. 13. Riker states that it is N!. William H. Riker, “Arrow's Theorem and Some Examples of the Paradox of Voting,”Mathematical Applications in Political Science, I, edited by John M. Claunch (Dallas: Southern Methodist University Press, 1965, p. 45. The following enumeration for N=3 is a counterexample to both statements: a1>a2>a3, a1>a3>a2, a2>a1>a3, a2>a3>a1, a3>a1>a2, a3>a2>a1, a1=a2>a3, a1=a3>a2, a2=a3>a1, a1>a2=a3, a2>a1=a3, a3>a1=a2, and a1=a2= a3. We conjecture that the correct number is the sum of all 2N-1 terms of the form (N1,\((N_1 ,\mathop {...}\limits^N N_R )\).
Riker, “The Paradox of Voting and Congressional Rules for Voting on Amendments,” p. 354.
For Lewis Carroll's and Dodgson's answers, see Thomas W. Casstevens, “The Caucus-Race: Teaching Cyclical Majorities,”P.S., 3, 1 (Winter, 1970), p. 27; for a third answer, see Riker, “The Paradox of Voting and Congressional Rules for Voting on Amendments,” pp. 364–6; for a fourth answer, see Black,The Theory of Committees and Elections, Ch. 9; for a fifth answer (“That alternative [among those in the cycle] which receives a plurality of the first place votes is declared the committee's choice.”), see Mark A. Satterthwaite, “Coalition Constructing Voting Procedures,” (Paper presented at the Annual Meeting of The Public Choice Society, 1972), pp. 18–19; for a discussion of possible criteria for an answer, see C. West Churchman,Prediction and Optimal Decision, Philosophical Issues of a Science of Values (Englewood Cliffs, N. J.: Prentice-Hall Inc., 1961), Ch. 12.
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Marz, R.H., Casstevens, T.W. & Casstevens, H.T. The hunting of the paradox. Public Choice 15, 97–102 (1973). https://doi.org/10.1007/BF01718845
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DOI: https://doi.org/10.1007/BF01718845