The theoretical potential for "voting cycles" was discovered by the French mathematician Condorcet in the eighteenth century (Mueller, 1989, p. 64) and received renewed attention in modern economics with the "impossibility theorem" of Arrow (1963). Suppose that three voters are to choose among three alternatives and the preferences of the individual voters are
Voter I[greater than]Z[greater than]Z Voter II Y[greater than]Z[greater than]X Voter III Z[greater than]X[greater than]Y
where "X[greater than]Y" means "alternative X is preferred to alternative Y." If these voters decide between two alternatives at a time by simple majority vote, both voters I and II will vote for Y rather than Z, and voters I and III will vote for X rather than Y. But voters II and III will vote for Z rather than X, so the collective choice is intransitive:
X[greater than]Y[greater than]Z[greater than]X
In this case majority rule results in a "voting cycle." There is no alternative which is a "Condorcet winner" that, when paired against the other alternatives, can defeat all of them by a majority vote.
The practical significance of voting cycles depends on the frequency of their occurrence. Some previous research on this topic has approached this question theoretically, exploring how the probability of cycles depends on the number of voters and the number of alternatives to be considered, assuming individual rankings of these alternatives are random (DeMeyer and Plott, 1970; Garman and Kamien, 1968; Niemi and Weisberg, 1968). Some of the major conclusions of this research are summarized in the next section.
Other studies have presented empirical evidence of the actual occurrence of a combination of individual preferences that would produce voting cycles. Dobra and Tullock (1981) examined a faculty search committee's rankings of 37 job candidates and found an example of "tie intransitivity" in which one candidate was not defeated by any other candidate but tied with three candidates, each of whom were beaten by other candidates beaten by the first candidate. Dobra (1983) reviewed 32 cases from other studies - many of which were also from academic settings - and found 3 tie-cycles and one complete cycle. He noted that "in most cases where a Condorcet winner existed the number of voters was large relative to the number of alternatives" (Dobra, 1983, p. 243).
This paper fits into the latter category of empirical studies. It presents a case study of the occurrence of voting cycles in business school curriculum reform. Three separate decisions are analyzed: (1) addition of a service requirement, (2) the inclusion of additional business courses in the core, and (3) changes in the nature of majors/"concentrations." A voting cycle was found in (2) but not (1) or (3).
II. THEORETICAL ANALYSES OF THE PROBABILITY OF VOTING CYCLE
Theoretical analysis may focus on the probability that pairwise majority voting results in a complete transitive ranking of all alternatives. Or the analysis may be limited to the probability that there is a "Condorcet winner," an alternative which wins a majority vote in pairwise comparisons with all other alternatives. In the case of three alternatives, a transitive ranking is equivalent to a Condorcet winner, but with more than three alternatives the distinction is necessary. A Condorcet winner may exist even if majority voting fails to produce a transitive ranking among all other alternatives. If the only concern is whether majority rule produces a unique winner, the existence of a voting cycle among other alternatives may present no problem.
Theoretical analyses of the relationship between the probability of cycles and the number of voters and alternatives to be considered have frequently assumed individual rankings of these alternatives are random, with all of the possible rankings equally likely (DeMeyer and Plott, 1970; Garman …