Voting cycles in business curriculum reform, a note.

Beck, John H.

I. INTRODUCTION

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 and Kamien, 1968; Niemi and Weisberg, 1968). Garman and Kamien (1968, pp. 312-313) refer to this equiprobability assumption as an "impartial culture." The formulas for calculating the probabilities of voting cycles are quite complex. Table 1 shows some representative values of the probability that there is no Condorcet winner with three voters, seven voters, and for the limiting case with an infinite number of voters. The probability of no Condorcet winner increases monotonically with the number of voters. Niemi and Weisberg (1968, p. 322) conclude:

. . . the maximum difference between exact and limiting values is very small for a small [number of alternatives]. Moreover, the exact values approach the limit rapidly. Thus, for a small number of alternatives, the number of individuals is almost totally irrelevant.

TABLE 1

Probabilities That There is no Condorcet Winner Under the Impartial
Culture Assumption

Number of Number of Voters
Alternatives 3 7 [infinity]

3 .05556 .07502 .08774
6 .20222 .27 .31524
8 .27075 .41509
15 .4175 .6087
[infinity] 1.0000

SOURCES: Garman and Kamien (1968, p. 314); Niemi and Weisberg
(1968, p. 322): DeMeyer and Plott (1970. p. 353)

On the other hand, as the number of alternatives increases the probability of no Condorcet winner increases more substantially, with the probability approaching one as both the number of alternatives and the number of voters approach infinity.

When the assumption that all possible individual rankings are equally probable is relaxed, the probability that there is no Condorcet winner may be either higher or lower than under the impartial culture assumption (Abrams, 1976 and 1980, p. 95). However, some authors have found a tendency for the probability that there is no Condorcet winner to decline with increases in various measures of "social homogeneity" indicating the extent to which voters have similar preferences (Fishburn and Gehrlein, 1980; Gehrlein and Fishburn, 1976; Mueller, 1989, p. 81).

III. A CASE STUDY OF VOTING CYCLES IN BUSINESS CURRICULUM REFORM

In 1994 the School of Business Administration at Gonzaga University engaged in a major reform of its curriculum. The analysis in this section is based on survey of faculty preferences about various curriculum alternatives. The survey was conducted solely for this research project after the curriculum reform process was completed, so there is no reason to suspect respondents might misrepresent their preferences to gain a strategic advantage.

Three separate decisions are analyzed: (1) addition of a community service requirement, (2) additions to the business core, and (3) changes in the nature of majors/"concentrations." Although community service has recently been adopted as part of the general education requirements at some colleges, there has been little discussion of its inclusion in business school curricula. The other two issues have been more widely considered by business schools. For example Dudley et al. (1995) advocate adding human resource management skills and upper level computer courses to the business core while reducing the number of credits in specializations.

The faculty preferences on curriculum reform expressed in the survey responses are shown in Tables 2-4. In these tables "X[greater than]Y" means "alternative X is preferred to alternative Y," and "X = Y" denotes indifference between X and Y. Note that allowing indifference departs from the common assumption in the theoretical analyses that individuals have strict preferences among all alternatives.

Table 2 shows preferences for 4 alternatives regarding a community service course, either as a requirement or as an option to satisfy an existing core requirement. The preferred choice of 7 faculty was not to include the service course in the core (alternative D). Three faculty preferred to require the service course, either in addition to (alternative A) or instead of (alternative C) an existing non-business requirement. The first choice of 10 faculty was to include the service course as an option to meet the Social Science requirement of the old core (alternative B). Thus alternative B would have been only one vote shy of a majority if all three alternatives had been considered together and individuals voted for their most-preferred alternative. It is therefore not surprising that in a series of pairwise comparisons simple majority voting produces a transitive ranking of all four alternatives with B as the first choice.

Table 3 shows preferences regarding three proposed additions to the business core. The 3-credit "World of Business" course would provide an integrative introduction to all the business disciplines. The 3-credit "Managerial Skills" course would emphasize practice developing "soft" skills like working in groups, etc., and the 2-credit "Advanced Business Computing" would deal with programming and simulation. Considering all of the possibilities of including or excluding these courses, there are 8 alternatives to be considered.

TABLE 2

Preferences for a Service Requirement

Individual
Preference Number
Orderings of Faculty

A[greater than]C[greater than]B[greater than]D 1
B[greater than]A[greater than]C[greater than]D 2
B[greater than]C[greater than]A[greater than]D 3
B[greater than]C[greater than]D[greater than]A 1
B[greater than]D[greater than]A[greater than]C 1
B[greater than]D[greater than]A=C 1
B[greater than]D[greater than]C[greater than]A 1
B = D[greater than]C[greater than]A 1
C[greater than]A[greater than]B[greater than]D 2
D[greater than]A=B=C 1
D[greater than]B[greater than]C[greater than]A 6

Majority Rule Ranking: B[greater than]D[greater than]C
[greater than]A

Definitions of Alternatives:

A Add PHI 271 Community Outreach (service) as an additional
requirement to the old SBA Core.

B Add PHI 271 Community Outreach (service) as an option to meet
the Social Science requirement of the old SBA Core.

C Add PHI 271 Community Outreach (service) as a requirement,
offsetting this by reducing the Social Science, Fine Arts,
History. or Science requirement in the old SBA Core by 3 credits.

D Do not have any service requirement in the core.

[TABULAR DATA FOR TABLE 3 OMITTED]

Table 3 shows that every one of the 20 faculty had a different ranking. In some cases, such as the first two individual preferences shown, the individual rankings differed only because one person expressed indifference between two alternatives where another expressed a strict preference. Still, the lack of similarity among individual preferences is striking, with 6 of the 8 alternatives being the first choice of at least one of the 20 faculty. With the larger number of alternatives and less social homogeneity in preferences, it may not be surprising that in this case there is no Condorcet winner. Simple majority voting produces a cycle: E[greater than]H = G[greater than]E. The three alternatives in this cycle - E, G, and H - all defeat any of the other 5 alternatives in pairwise votes.

Table 4 shows preferences regarding how much coursework students should be required to take in one of the traditional business disciples and whether majors would be designated in these disciplines or whether all students would [TABULAR DATA FOR TABLE 4 OMITTED] have a "Business Administration major" with the traditional disciplines only designated as "concentrations." Note that although 6 alternatives are listed, these represent only 3 different alternatives in terms of the substantive requirements for coursework; the other differences among the alternatives are only the labels "major" and "concentration." Table 4 shows that labels did matter. Except for the first individual, no one was indifferent between the alternatives that were substantively identical in terms of course requirements. Every one of the 6 alternatives was the first choice of at least one individual, and only two individuals shared identical preferences. However, in this case the diversity of individual preferences did not produce a voting cycle. Simple majority rule produces a transitive ranking of all 6 alternatives: Q[greater than]N[greater than]O[greater than]P[greater than]R[greater than]M.

IV. DISCUSSION

When individual preferences are such that a potential voting cycle exits, the outcome depends on the order in which pairs of alternatives are considered. Thus, the person controlling the agenda could secure his preferred outcome by determining the sequence in which alternatives were considered (Riker, 1982, p. 137). Consider again the example from the introduction with three voters with the following preferences:

Voter I X[greater than]Y[greater than]Z Voter II Y[greater than]Z[greater than]X Voter III Z[greater than]X[greater than]Y

If Voter I controls the agenda, he can secure his preferred alternative X by having the group consider alternatives Y and Z first and then proposing X as an alternative to Y, the winner of the first vote.

There is also the possibility that individuals might behave strategically, not voting sincerely but at some stages of the process voting against the alternative in a pair which they honestly prefer in the hope that by eliminating this alternative from further consideration they will secure a better outcome at the end of the process (Riker, 1982, p. 137). Referring again to the same example, if Y and Z are considered first, Voter II might vote for Z rather than Y. Then, when Voter I proposes X as an alternative to the winner of the first vote, Z will defeat X and Voter II gets his second choice Z rather than his last choice X.

The effectiveness of agenda control or insincere voting depends on individuals' knowledge of the preferences of other voters. My impression of the curriculum reform at Gonzaga University is that the faculty had no good idea of their colleagues' preferences. Indeed, at one point, advocates of radical changes made more concessions than necessary to win support for at least some of the changes that they favored. Thus, I do not think that deliberate manipulation played a role in the outcome. Dobra (1983, p. 246) also noted the apparent absence of attempts to manipulate the choice of a personnel selection committee when individual rankings might have created a voting cycle.

Associate Professor of Economics, School of Business Administration, Gonzaga University, Spokane, WA 99258. (509)328-4220 ext. 3429. E-mail: BECK@JEPSON.GONZAGA.EDU I am grateful to Clark Wiseman for comments on an earlier draft of this paper.

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