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Paradoxes of Voting*

Published online by Cambridge University Press:  01 August 2014

Peter C. Fishburn
Peter C. Fishburn*
Affiliation:
The Pennsylvania State University
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Abstract

Five voting paradoxes are examined under procedures which determine social choice from voters' preference rankings. The most extreme forms of each paradox are identified, and their potential practical significance is assessed using randomly generated voter preference profiles.

The first paradox arises when the winner under sequential-elimination simple-majority voting is less preferred by every voter than some other alternative. The fifth paradox occurs when one alternative has a simple majority over every other alternative and one or more of the simple-majority losers beats the winner on the basis of every point-total method that assigns more points to a first-place vote than to a second-place vote, more points to a second-place vote than to a third-place vote, and so forth.

The other three paradoxes are solely concerned with point-total procedures. They include cases in which the standard point-total winner becomes a loser when original losers are removed, and in which different truncated point-total procedures (which count only first-place votes, or only first-place and second-place votes, and so forth) yield different winners.

The computer simulation data suggest that the more extreme forms of the paradoxes are exceedingly unlikely to arise in practice.

Type
Articles
Information
American Political Science Review , Volume 68 , Issue 2 , June 1974 , pp. 537 - 546
Copyright
Copyright © American Political Science Association 1974

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Footnotes

*

I wish to thank Bruce McCarl for his assistance in writing the computer program used in the simulation analyses of this paper.

References

1 This seems especially true when no alternative has a simple majority over every other alternative (Condorcet's famous paradox of voting). I shall argue later, however, that, even when there is a simple-majority winner, some other alternative might seem more attractive as the social choice.

2 de Borda, Jean-Charles, “Mémoire sur les élections au scrutin,” Histoire de l'Académie Royale des Sciences, 1781 Google Scholar (translated, with commentary, by de Grazia, Alfred, “Mathematical Derivation of an Election System,” Isis, 44 [06 1953], 42–51 CrossRefGoogle Scholar); de Condorcet, Marquis, Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix, Paris, 1785 Google Scholar. See Black, Duncan, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958) for discussion of these and other historical figuresGoogle Scholar.

3 Copeland, A. H., “A ‘Reasonable’ Social Welfare Function,” mimeographed, University of Michigan Seminar on Applications of Mathematics to the Social Sciences, 1951 Google Scholar.

4 Fishburn, Peter C., “A Comparative Analysis of Group Decision Methods,” Behavioral Science, 16 (11 1971), 538–544 CrossRefGoogle Scholar; The Theory of Social Choice (Princeton: Princeton University Press, 1973)Google Scholar; “Subset Choice Conditions and the Computation of Social Choice Sets,” Quarterly Journal of Economics, forthcoming.

5 Some key references that will lead to others are: Riker, William H., “Voting and the Summation of Preferences: An Interpretive Bibliographical Review of Selected Developments During the Last Decade,” American Political Science Review, 55 (12 1961), 900–911 CrossRefGoogle Scholar; May, Robert M., “Some Mathematical Remarks on the Paradox of Voting,” Behavioral Science, 16 (03 1971), 143–151 CrossRefGoogle Scholar; and Part 3 in Probability Models of Collective Decision Making, ed. Niemi, Richard G. and Weisberg, Herbert F. (Columbus, Ohio: Charles E. Merrill, 1972)Google Scholar.

6 It is tempting to suggest that the more surprising the paradox, the less likely it is to occur in practice. This of course is a matter of personal judgment that the reader may wish to assess for himself.

7 Note that in C(x 1 x 2 … xm ), x 1 has an m − 1 to 1 majority over x 2, x 2 has an m − 1 to 1 majority over x 2, …, and xm has an m − 1 to 1 majority over x 1. The same is true for C′ since its orders are obtained from a permutation of the orders in C.

8 Harary, Frank, Norman, Robert Z. and Cartwright, Dorwin, Structural Models: Art Introduction to the Theory of Directed Graphs (New York: Wiley, 1965), 314 Google Scholar.

9 Fishburn, Peter C., “The Irrationality of Transitivity in Social Choice,” Behavioral Science, 15 (03, 1970), 119–123, esp. p. 122 CrossRefGoogle Scholar; Fishburn, , “A Comparative Analysis of Group Decision Methods,” esp. p. 542 Google Scholar.

10 Davidson, Roger R. and Odeh, Robert E., “Some Inconsistencies in Judging Problems,” Journal of Combinatorial Theory (A), 13 (1972), 162–169 CrossRefGoogle Scholar.

11 Arrow, Kenneth J., Social Choice and Individual Values, 2nd ed. (New York: Wiley, 1963), pp. 26–27 Google Scholar.

12 For further discussion on this point see Peter C. Fishburn, “The Irrationality of Transitivity in Social Choice,” and “Should Social Choice be Based on Binary Comparisons?,” Journal of Mathematical Sociology, 1 (01 1971), 133–142 CrossRefGoogle Scholar.

13 Peter C. Fishburn, “On the Sum-of-Ranks Winner when Losers are Removed,” Discrete Mathematics, forthcoming.

14 Black, Duncan, Theory of Committees and Elections, 176–177 Google Scholar.

15 For a proof of Theorem 5 see Theorems 6.1 and 7.1 in Fishburn, Peter C., Decision and Value Theory (New York: Wiley, 1964)Google Scholar.

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