THE OCCURRENCE OF THE PARADOX OF VOTING 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 PUBLIC CHOICE 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 ballot 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 single transferable vote. 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.