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Majority Decision-Making with Partial Unidimensionality*

Published online by Cambridge University Press:  01 August 2014

Richard G. Niemi
Richard G. Niemi*
Affiliation:
University of Rochester
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Extract

A major dilemma for majority decision-making occurs when the summation of transitive individual preference orderings results in an intransitive social ordering. The problems posed by this phenomenon, which is known as the paradox of voting, can be seen in the following standard example.

Suppose there are three individuals, one with each of the following preference orders of three alternatives: ABC, BCA, CAB. Under majority rule, A would defeat B, B would defeat C, and C would defeat A, so there is no majority winner.

Most voting procedures, of course, yield a unique result whether or not the paradox occurs. But from this example it is apparent that when the paradox does occur, a majority of the voters prefer an alternative other than the one which is selected. Moreover, if a typical voting procedure is used, which of the alternatives is selected depends on the order in which the alternatives are voted on. Clearly these results have important implications, whether one is concerned with normative questions about majority rule or with the practical politics of legislative decision-making.

In the burgeoning literature on the voting paradox, surely one of the most impressive and well-known findings is Black's and Arrow's demonstration that the paradox cannot occur if the set of individual preference orderings is single-peaked. Since single-peakedness implies that the individuals and alternatives can be arrayed on a single dimension, their finding has a meaningful substantive interpretation. Namely, complete agreement on a dimension for judging the alternatives ensures that majority voting will yield a transitive social ordering of the alternatives.

Type
Research Article
Information
American Political Science Review , Volume 63 , Issue 2 , June 1969 , pp. 488 - 497
Copyright
Copyright © American Political Science Association 1969

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Footnotes

*

I would like to thank Herbert Weisberg, whose continued collaboration helped stimulate this research. I also wish to thank James Callan for technical help. The University of Rochester Computing Center provided computer time.

References

1 Black, Duncan, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958), pp. 39–40 Google Scholar.

2 Black, op. cit., Chap. IV; Arrow, Kenneth J., Social Choice and Individual Values, 2nd ed. (New York: Wiley, 1963), pp. 74–80 Google Scholar. A set of preference orderings is single-peaked if there is an ordering of the alternatives on the abscissa such that when utility or degree of preference is indicated by the ordinate, each preference ordering can be represented by a curve which changes its direction at most once, from up to down (i.e. has at most one peak).

3 For most political scientists, the major work relying on the assumption of unidimensionality is Downs, Anthony, An Economic Theory of Democracy (New York: Harper, 1957)Google Scholar. In Downs' spatial analysis voters and parties unanimously view the electoral struggle in terms of a single ideological dimension. They do not, of course, agree on which party is best, but they do share a common dimension or continuum on which to judge the alternatives. Downs' work has often been criticized because unidimensionality is, assumed. In a sense, I am making the same criticism of one aspect of Black's and Arrow's work, but I am also proceeding (a little way) with the analysis after dropping the criticized assumption.

4 More general criteria than single-peakedness will also prevent the occurrence of the paradox. However, as with single-peakedness, every individual's preference ordering must satisfy the criteria. In addition, the more general conditions lack a meaningful substantive interpretation. See Sen, Amartya K., “A Possibility Theorem on Majority Decisions,” Econometrica, 34 (1966), 491–499 CrossRefGoogle Scholar, and the references cited therein.

5 Coombs, Clyde H., A Theory of Data (New York: Wiley, 1964), Chaps. 5–7Google Scholar.

6 Coombs distinguishes a quantitative J scale, on which the spacing as well as the order of alternatives is determined, from a qualitative J scale, on which the order of alternatives but not their spacing has been determined. We will only be concerned with the latter type.

7 Coombs, op. cit., p. 395.

8 Niemi, Richard G. and Weisberg, Herbert F., “A Mathematical Solution for the Probability of the Paradox of Voting,” Behavioral Science, 13 (1968), 317–323 CrossRefGoogle ScholarPubMed.

9 In general, given m alternatives, the minimum proportion of I scales satisfying a common qualitative J scale is [2n–1/n!]. To show this requires a more general proof, which is not useful for the present analysis.

10 Probabilities of the paradox for various numbers of individuals and alternatives are given in Campbell, Colin D. and Tullock, Gordon, “A Measure of the Importance of Cyclical Majorities,” The Economic Journal, 75 (1965), 853–856 CrossRefGoogle Scholar; Frank Demeyer, and Charles R. Plott, “The Probability of a Cyclical Majority,” Econometrica, forthcoming; Garman, Mark B., and Kamien, Morton I., “The Paradox of Voting: Probability Calculations,” Behavioral Science 13 (1968), 306–316 CrossRefGoogle ScholarPubMed; Klahr, David, “A Computer Simulation of the Paradox of Voting,” this Review, 60 (1966), 384–390 Google Scholar; Niemi and Weisberg, op. cit.; Pomeranz, John E. and Weil, Roman L. Jr., “Calculation of Cyclical Majority Probabilities,” unpublished paper, University of Chicago Google Scholar.

11 In our earlier paper, questions were raised about the interpretation of the initial probability assumptions. These questions are fully applicable here. See Niemi and Weisberg, op. cit., fn. 6. It should be added that instances in which the paradox is contrived are not adequately accounted for by a probabilistic model.

12 Niemi and Weisberg, op. cit.

13 One can find the total probability of the paradox (within rounding error) by multiplying the corresponding terms in Tables 1 and 2. For example, for m=3 , .250 (.222)+0(.778) =.0555.

14 David, Paul T., “Experimental Approaches to Vote-Counting Theory in Nominating Choice,” this Review, 56 (1962) 673–676 Google Scholar.

15 The smaller set obviously cannot contain more than half of the preference orderings which satisfy the common J scale. On the other hand, suppose it contained only m–x–y I scales. Then the other two sets satisfy a common J scale, and they contain m – (m–x–y)=x+y individuals. But this is contrary to the assumption that only x preference orderings satisfy a common J scale.

16 Tables of the Binomial Probability Distribution, Appl. Math. Series 6 (Washington: U. S. Government Printing Office, 1949)Google Scholar.

17 Tables of the Cumulative Binomial Probability Distribution (Cambridge: Harvard University Press, 1955)Google Scholar. Romig, Harry G., 50–100 Bimonial Tables (New York: Wiley, 1953)Google Scholar.

18 The proof in this paragraph—i.e. when exactly 2/3 of the preference orderings satisfy a common J scale—holds for every value of m which is a multiple of three (and an odd number).

19 Feller, William, An Introduction to Probability Theory and Its Applications, 2nd ed., Vol. 1 (New York: Wiley, 1957), p. 141 Google Scholar.