The Probability of Condorcet Cycles and Super Majority Rules

Abstract Majority voting aggregates individual preference profiles into a binary relation on the set of alternatives. Condorcet cycles are cycles of the aggregated binary relation. We show that the relative volume of the subset of the (n!−1)-simplex that represents profile distributions such that the aggregated preferences display Condorcet cycles is a decreasing function of the super majority levelτbounded by the expression n! 1−τ 0.4714 n! . This expression shows that Condorcet cycles become rare events for super majority rules larger than 53%.Journal of Economic LiteratureClassification Number: D71.

[1]  D. Saari.  Geometry of voting , 1994 .

[2]  Andrew Caplin,et al.  ON 64%-MAJORITY RULE , 1988 .

[3]  K. Arrow,et al.  Social Choice and Multicriterion Decision-Making , 1986 .

[4]  N. Schofield.  The Geometry of Voting , 1983 .

[5]  Peter C. Fishburn,et al.  Condorcet proportions and Kelly's conjectures , 1979, Discret. Appl. Math..

[6]  Joseph Greenberg,et al.  CONSISTENT MAJORITY RULES OVER COMPACT SETS OF ALTERNATIVES , 1979 .

[7]  Jean-Michel Grandmont,et al.  INTERMEDIATE PREFERENCES AND THE MAJORITY RULE , 1978 .

[8]  Peter C. Fishburn,et al.  The probability of the paradox of voting: A computable solution , 1976 .

[9]  Jerry S. Kelly,et al.  Voting Anomalies, the Number of Voters, and the Number of Alternatives , 1974 .

[10]  Donald E. Knuth,et al.  The Art of Computer Programming, Vol. 3: Sorting and Searching , 1974 .

[11]  S. Tanny.  A probabilistic interpretation of Eulerian numbers , 1973 .

[12]  Gerald H. Kramer,et al.  On a Class of Equilibrium Conditions for Majority Rule , 1973 .

[13]  Robert M. May,et al.  Some mathematical remarks on the paradox of voting , 1971 .

[14]  C. Plott,et al.  The Probability of a Cyclical Majority , 1970 .

[15]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[16]  Charles R. Plott,et al.  A Notion of Equilibrium and Its Possibility Under Majority Rule , 1967 .

[17]  Gordon Tullock,et al.  The General Irrelevance of the General Impossibility Theorem , 1967 .

[18]  G. Thompson,et al.  The Theory of Committees and Elections. , 1959 .

[19]  J. Riordan.  Introduction to Combinatorial Analysis , 1958 .

[20]  L. A. Goodman,et al.  Social Choice and Individual Values , 1951 .