THE OCCURRENCE OF THE OF IN UNIVERSITY ELECTIONS

Richard G. Niemi*

Despite its theoretical importance for both normative and descriptive concerns, political scientists have largely ignored the paradox of voting because of the lack of solid evidence that it ever occurs in practice. Methods have been found to tell us the expected probability of the paradox if certain assumptions are made about individual preference orderings [1, 2, 3, 4, 5, 6]. But how closely these assumptions (and hence the resulting probabilities) correspond to reality is not known. Moreover, voting procedures used in most institutional settings provide insufficient information to determine whether or not the paradox was present. Only by a careful analysis of the situation can one even infer that a voting cycle existed [7, 8]. In fact, I am aware of only one reported instance in which the occurrence of the paradox is documented on the basis of complete individual preference orderings [ 10, pp. 45-46].

In order to change this situation, I report here the frequency of the paradox in a series of actual elections held by a University faculty. Since voters in these elections were asked to rank order the candidates, it was possible to determine definitely whether the paradox occurred. It was also possible to examine the individual preference orderings for some insight into when and why the paradox exists. While there are too few elections to provide us with a firm estimate of the empirical frequency of the paradox, the analysis shows that cyclical majorities are by no means absent in real world voting, and suggests that the paradox may indeed be more than an interesting curiosity.

I. THE PARADOX OF VOTING

Good descriptions of the voting paradox are found in a number of places (especially [8], pp. 900-901), and only a brief sketch of the problem is required here. We begin with n alternatives, A 1 ..... An, which are ranked by a number of individuals. The following assumptions are made:

I. For each individual, the rank Order is transitive; that is, for all i, j, k, if A i • ~ Aj (read: A i is ranked above Aj) and Aj" ~" Ak, then Ai'~A k.

II. In voting on a pair of alternatives, each individual votes for the alternative which stands higher in his rank order.

The author is Assistant Professor of Politica! Science at the University of Rochester. I would like to thank Professor Herbert Weisberg for his helpful comments on an earlier version of this paper. 92

Using these assumptions along with the individual rank orders, comparisons can be made between each pair of alternatives in order to arrive at a social ordering of the alternatives. The following standard example illustrates the procedure and also shows how the paradox can occur:

Let there be three individuals, one with each of the following rank orders of three alternatives: AIA2A3, A2A3A1, A3AIA 2. By majority rule, we find that for this set of individuals A 1 • >A2, A 2 • >A3, and A 3 • >A 1.

This result- that the summation of transitive individual rank orders sometimes results in an intransitive social ordering - is called the paradox of voting.

Theoretical work on the paradox often makes use of two additional simplifying assumptions- that the number of individuals is odd and that indifference between alternatives is not permitted. In empirical analyses these assumptions must be discarded. Even numbers of voters create no basic problems, although they do allow "ties" to exist. This happens in the votes analyzed below, and will be dealt with at that point. Failure to rank some of the alternatives (interpreted as indifference) occurs very often in the votes used here, especially in three elections with large numbers of candidates. In the analysis of pairwise preferences below, it is assumed that a voter was indifferent between two candidates if neither of them was ranked. If only one was ranked, this one was taken as preferred. For example, if there were five candidates A,..., E, and a voter indicated that his preference ordering was B • ~A • >E, it was assumed that C and D were tied with each other and that both of them ranked below B, A, and E.

II. THE VOTING PARADOX IN UNIVERSITY ELECTIONS

Using the above procedures, rank orders of the candidates from 22 elections were examined in order to determine whether the paradox occurred. The elections took place over a four year period at the University of Rochester. They include all of the elections in which three or more candidates were nominated for the University Senate or for a university committee. (There were some 40 elections altogether.) For four elections the electorate was the entire University faculty. One election was limited to the School of Medicine and Dentistry, and the remainder took place within the College of Arts and Science.

In each election the instructed the voters to rank in order of preference anywhere from one to all of the candidates. It was also indicated that regardless of how many candidates a voter ordered, ranking one candidate over some other(s) might help that candidate's chance of being elected, and could never reduce his chances. Votes were to be counted by the Hare system with a . As indicated below, most of the elections had from three to six candidates for PARADOX OF VOTING 93 one or two positions. In the Senate elections, however, 15 men were elected from candidate lists of 22 to 36 nominees.

The social ordering for each election, along with collateral information, are presented in Table 1. In the 18 elections with six or fewer candidates, intransitive orderings occurred four times, once involving a complete cycle (Election 6) and three times involving ties (Elections 2, 14 and 15). 1 In the election with five candidates, and by necessity in those with three, the intransitivity was such that no candidate would have received a majority over every other one in pairwise competition.

The remaining elections are more complicated because of the large numbers of candidates involved. Actually, the social orderings in Elections 19 and 20 are almost completely transitive, involving only two ties and one cycle, respectively. In the last two elections some 35 ties or cycles exist, although in each case none of them involved the top five candidates.

In a number of elections (2, 6, 9, 11, 12, 15, 16, 19-22), both those in which the paradox did and did not occur, the candidate(s) elected would not have defeated all of the losers by a majority vote in pairwise competition. This can happen in almost all electoral systems and in itself should not be surprising (although the number of times it happened in these elections may be disconcerting). What is important is that when the paradox occurred, no matter which candidate was elected, he could have been defeated (or at least tied) by another candidate. This is, of course, the essence of the paradox.

Discussion

The elections analyzed here are too few and of too limited a variety to be a reliable indicator of the frequency of the paradox in voting situations generally. Nor can they be compared in detail with theoretical frequencies generated on the basis of convenient assumptions about the individual rank orders. Nevertheless, several seemingly important points can be made. Their real significance awaits observation and analysis of other voting records.

The first point that can be made is that two of the assumptions usually associated with the theoretical model of the paradox are untenable for this set of elections. Of course some departure from these assumptions should be expected in

1While such relationships as A • ~B, B - >C, C=A (read: C is tied with A) are clearly intransitive, it is a semantic question whether they are instances of the paradox or whether the paradox includes only complete voting "cycles." Since in either case no alternative can defeat both of the other alternatives (without some tie-breaking mechanism), I will speak of both "ties" and "cycles" as instances of the paradox. 94 PUBLIC CHOICE

Table 1 The Social Ordering of Candidates in Twenty-Two University Elections

Candi- Election Number of Number to Number date(s) Number Candiates be Elected of Voters Social Ordering a Elected

1 3 1 106 A • ~B • >c A

2 3 1 114 A .~B • ~C; C=A A

3 3 1 106 A - >B >c A

4 3 1 88 A - >B >C A

5 3 1 93 A - >B >C A

6 3 1 100 A • ~S >c; C->A ?~

7 3 1 81 A - ~B >c A

8 4 1 126 A • ~B >C" >D A

9 4 1 105 A • >B >C • >D B

10 5 2 115 A . ~B • >... ">E A,B

11 5 2 117 A -~B >....>E B,c

12 5 2 90 A - ~B • >...->E A,C

13 5 2 106 A • ~B • >...->E A,B

14 5 2 102 A • >B • ~>..." >E; A,B

D=A

15 5 2 96 A • >B=C ">D'>E; A,B

C=A

16 5 3 90 A .>B .>....>E A,B,D

17 6 1 134 A . >B . >. . . >F A

18 6 1 123 A->B "•..'>F A

19 22 t5 453 A.>B.>. ..>v; A-J,

I=G, L=J L-O, R

20 26 15 390 A.>B.>...>z; A-I, K,

V • ~S M,O-R

21 b 35 15 428 A" >B" >..* >Z A-H,

• >AA.~>. .>tl; L-N, P,

H .>F, K->H, S,T,V PARADOX OF VOTING 95

L'>,G, L ">H,

L.>J,O.>G,

O'>K,O ">M, o.>K,°'>M,

O. >o, T. >Q, U .>P, V->S,

Z'>W, AA'>R,

AA'>S, AA'>V,

BB " >V, BB=W,

CC " ~X, CC=AA,

GG=EE, tl •>GG

22 b 36 15 463 A->B->...->Z A-D,

• >AA'>...'>JJ; F-H,

H=F, J ' >G, K-M, K, >F, L ->J, P, U-X

P'~>M, R=O,

U ' ~S, DD=AA,

EE • ~>AA, EE=CC,

HH=FF

a The symbol ", ~" means "is ranked above." The "='" sign means "is tied with." All relationships are transitive unless otherwise noted• Letters are assigned to candidates so that the winner is A, the second-place candidate is B, etc.

b The social ordering for these elections cannot be represented uniquely. In several cases the order of the candidates could be changed with no increase in the number of intransitivities.

c After this election was conducted, the committee was disbanded in a reorganization move.

any set of empirical data. But the magnitude of the deviations in these votes is cor~siderable. Perhaps the main departure is from the assumption that indifference among the alternatives is not allowed. In all of the University elections, 96 PUBLIC CHOICE indifference, as indicated by partial rank orders, was very much in evidence. The percentages of voters with varying degrees of indifference are shown in Table 2 for each election. It is not surprising, of course, that complete rankings become scarcer as the number of alternatives grows. Other things being equal, it is more work to rank six candidates than three candidates. But even when there are only three or four candidates the number of partial orderings is quite large. And ranking only one or two out of five or six nominees is a frequent occurrence. The effect of such large numbers of partial rankings on the likelihood of the paradox has not been investigated. But if these elections are a good indication, partial preference orderings are an important consideration.

Table 2 Percentages of Partial and Complete Preference Orderings in the University Elections

Election Number of Percentages of voters ranking the following number of candidates: Number Candidates 1 2 3 4 5 6 7-9 10-12 >~--13

1 3 31 a 69 2 3 26 -- 74 3 3 29 -- 71 4 3 14 -- 86 5 3 16 -- 84 6 3 21 -- 79 7 3 20 -- 80 1-7 3 23 -- 77 8 4 19 3 a 78 9 4 15 30 -- 55 8-9 4 17 15 -- 68 10 5 15 32 15 a 38 11 5 16 27 21 -- 36 12 5 13 26 13 -- 48 13 5 13 16 9 -- 61 14 5 18 14 14 -- 55 15 5 17 17 22 -- 45 16 5 10 7 28 -- 56 10-16 5 15 20 17 -- 48 17 6 5 22 29 11 a 33 18 6 12 23 17 14 -- 34 17-18 6 9 22 23 12 -- 33 19 22 6 15 16 18 13 8 14 4 5 20 26 5 6 21 15 12 13 13 8 7 21 35 2 4 10 14 15 7 22 17 9 22 36 5 10 20 16 13 10 14 7 6 19-22 22-56 4 9 17 16 13 9 16 9 7

aThese cells are empty by definition, since if the first n=l preferences are indicated, the remaining alternative is the least preferred. PARADOX OF VOTING 97

The other common assumption made in theoretical work on the paradox is that voters have an equal probability of possessing any of the n! complete rank orders, where n is the number of alternatives. (This is called the equally-likely assumption.) If this assumption were correct, we would expect about l/n! voters with each rank order. Using only the complete preference orderings for the elections with three or four candidates, it is apparent that the assumption is far from correct in these cases. 2 While it is easy to reject the assumption of equally-likely rank orders, however, it is by no means immediately apparent what assumption(s) ought to be made. In lieu of trying to justify some other assumption on the basis of these few votes, we simply present the distributions of complete rank orders in Figure 1, along with a hypothetical distribution in which the voters were divided equally among the rank orders. (For the three-candidate elections, two figures are drawn simply to improve readability.)

We have thus far shown that the preferences in the University elections Vary rather widely from the assumptions commonly underlying theoretical work on the paradox. Does this have any consequences for the frequency of its appearance? Only additional evidence can satisfactorily answer this question. It is worthwhile, however, to make a cursory comparison of theoretical expectations and this initial evidence. One comparison finds the empirical results directly contrary to theoretical expectations. In the University elections the absence of a majority winner occurred most frequently when there was a small number of candidates. In contrast, calculations based on the equally-likely assumption yield the following increasing probabilities of no majority winner for three, four, five, and six alternatives, respectively: 3 .088, .175, .251, .315. Moreover, on the basis of the same model, the likelihood of no majority winner in Elections 19, 20, 21, and 22 is very high: .703, .738, .791, and .796 respectively. Yet a majority winner occurred all three times. 4 Thus the empirical findings, if verified by future observations, suggest that the results based on the equally-likely assumption misrepresent reality in a crucial way.

On the second comparison with theoretical expectations, the little evidence that exists is ambiguous. It has been shown, for the three alternative case, that the

2Using a chi-square test of significance, the equally-likely assumption is rejected at the .05 level in both distributions of four alternatives and in five of the seven distributions of three alternatives. With five or more alternatives, nl is larger than the number of voters, so the test would be rather unfair,

3The values given here are for an infinite number of voters, The approach to the limit is fast enough that these are h,ighly satisfactory approximations of the real values (given the numbers of voters in the University elections).

4If the equally-likely model applied, the probability of a majority winner occurring in all four elections would be (.297) (.262) (.209) (°204) = .003. 98 PUBLIC CHOICE

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(/~lUOSSUueO~O • ~ du~o~ UO pgsDql PARADOX OF VOTING 99 theoretical probability of the paradox is inversely related to the amount of single-peakedness or unidimensionality in the individual orderings [4]. To compare this result with those in the empirical data, we calculated the degree of single-peakedness for the seven elections with three alternatives, using only the complete rank orders. For elections 1-7, respectively, the following proportions of complete rank orders satisfy single-peaked criteria: 74%, 80%, 87%, 75%, 82%, 72%, 83%. It is interesting to observe that, in line with theoretical expectations, the one complete voting cycle in these elections occurred when the fewest preference orderings were single-peaked. However, the tie in Election 2 occurred even though a rather large proportion of the orderings was single-peaked. 5 Only further studies will show whether this theoretical expectation is borne out empirically. They will also show how likely varying degrees of unidimensionality are in a variety of voting situations.

Ilt. CONCLUSION Twenty-two votes in a series of University elections hardly permit us to give a reliable estimate of the empirical frequency of the paradox of voting. More important than the small number of cases involved is the fact that these elections represent only a small part of the wide array of voting situations and procedures. Voting on issues in deliberative bodies might be considered a totally different type of situation than that considered here. Elections in which factions or parties compete involve factors not covered in this paper. The intrusion of strategic considerations, such as contrived occurrences of the paradox or rules which prevent the introduction of some alternatives, introduces additional complexities. Thus it would be presumptuous to offer an estimate of the frequency with which the paradox appears.

We can lay claim, however, to having shown that the paradox does occur in real world voting. Although more evidence is clearly needed before the paradox can be called a crucial (or perhaps even a minor) political phenomenon, presentation of the current findings and of other probable instances of the paradox [9, 10] may at the very least justify the search for more definitive data. Hopefully, it wilt also breathe a little substance into what many have heretofore been considered a mathematical irrelevancy.

5Under the theoretical model used, and with the number of individuals involved in the University elections, a shift from 72% to 80% single-peaked preferences considerably lowers the likelihood of the paradox, 100 PUBLIC CHOICE

REFERENCES

1. Demeyer, Frank, and Charles R. Ptott. "The Probability of a Cyclical Majority," Econometrica, forthcoming.

2. Garman, Mark B., and Morton I. Kamien. "The Paradox of Voting: Probability Calculations," Beh. Sci., 13 (1968), 306-316.

3. Gleser, Leon Jay, "The Paradox of Voting: Some Probabilistic Results," Public Choice 7 (1969), 47-63.

4. Niemi, Richard G., "Majority Decision-Making with Partial Unidimension- ality," Am. Pol. Sci. Rev. 63 (1969), 488-497.

5. Niemi, Richard G. and Herbert F. Weisberg. "A Mathematical Solution for the Probability of the Paradox of Voting," Beh. Sci. 13 (1968), 317-323.

6. Pomeranz, John E. and Roman L. Weil, Jr. "Calculation of Cyclical Majority Probabilities," unpublished paper, University of Chicago, 1968.

7. Riker, William H. "The Paradox of Voting and Congressional Rules for Voting on Amendments," Am. Pol. Sci. Rev. 52 (1958), 349-366.

8. Riker, William H. "Voting and the Summation of Preferences: An Interpretive Bibliographic Review of Selected Developments During the Last Decade," Am. Pol. Sci. Rev. 55 (1961), 900-911.

9. Riker, William H. "Arrow's Theorem and Some Examples of the Paradox of Voting," edited by John M. Claunch, Mathematical Applications in Political Science. Dallas: Arnold Foundation, 1965.

10. Taylor, Michael. "Graph-Theoretic Approaches to the Theory of Social Choice," Public Choice 4 (1968), 35-47.