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Vote Trading and the Voting Paradox: A Proof of Logical Equivalence*

Published online by Cambridge University Press:  01 August 2014

David H. Koehler
David H. Koehler*
Affiliation:
American University
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Abstract

Riker and Brams have demonstrated the paradox of vote trading (“… that rational trades by all members [may] make everyone worse off”). In so doing the authors indicate the existence of an apparent disequilibrium when vote trading occurs. I extend this latter point and prove that the preference conditions required for vote trading are the same as those which produce the cyclical majority; the conditions for vote trading and the cyclical majority are logically equivalent. The conclusion briefly indicates the impact of this finding with respect to the work of a number of other authors and gives some idea of the restrictions which would be required to eliminate vote trading among rational legislators.

Type
Vote Trading and the Voting Paradox: A Proof of Logical Equivalence
Information
American Political Science Review , Volume 69 , Issue 3 , September 1975 , pp. 954 - 960
Copyright
Copyright © American Political Science Association 1975

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Footnotes

*

I wish to extend my gratitude to the many individuals who read and commented on this work. Among them I owe a particular debt to Professor Steven J. Brams for his detailed criticism of several drafts and his continued encouragement.

References

1 Riker, William H. and Brams, Steven J., “The Paradox of Vote Trading,” American Political Science Review, 67 (12, 1973), 1241 CrossRefGoogle Scholar.

2 Riker and Brams, p. 1246.

3 Park, R. E., “The Possibility of a Social Welfare Function: Comment,” American Economic Review, 57 (12, 1967), 1304 Google Scholar.

4 Tullock, Gordon, “A Simple Algebraic Logrolling Model,” American Economic Review, 60 (06, 1970), 422 Google Scholar.

5 Ibid., p. 424.

6 Tullock makes the point in an appendix to Buchanan, James and Tullock, Gordon, The Calculus of Consent (Ann Arbor: University of Michigan, 1962), p. 332 CrossRefGoogle Scholar, as does Coleman, James, “The Possibility of a Social Welfare Function,” American Economic Review, 56 (1966), 1105–22Google Scholar. These remarks are addressed to Arrow, Kenneth J., Social Choice and Individual Values, Cowles Foundation Monograph No. 12, 2nd ed. (New York: Wiley, 1965), pp. 1–33 Google Scholar.

7 Since the completion of this work, several other papers dealing with the subject have come to my attention. Bernholz, Peter, “Logrolling, Arrow Paradox and Cyclical Majorities,” Public Choice, 16 (Summer, 1973), 87–102 CrossRefGoogle Scholar and “Logrolling, Arrow Paradox and Decision Rules: A Generalization,” Kyklos, 27 (11, 1973), 49–62 Google Scholar, has demonstrated that vote trading implies the voting paradox. He does not prove that they are logically equivalent. Also Kadane, Joseph, “On Division of the Question,” Public Choice, 13 (Fall, 1972) 47–54 CrossRefGoogle Scholar, and Miller, Nicholas R., “Logrolling and the Arrow Paradox: A Note,” unpublished manuscript (12, 1973)Google Scholar have addressed the equivalence by considering the situation where there is no undominated platform confronting a set of voters. This implies that members will have an incentive to trade votes, and that a cyclical majority is present.

Joe A. Oppenheimer, in “Relating Coalitions of Minorities to the Voters' Paradox or Putting the Fly in the Democratic Pie,” a paper delivered at the Annual Meeting at the South West Political Science Association Meeting, San Antonio, Texas, March 30—April 1, 1972, has shown that in legislatures or elections “an efficacious coalition of minorities can exist if, and only if, there is an underlying voters' paradox” (p. 5). Oppenheimer's logic is similar to that presented here; the structure of the two arguments, however, is quite different. If the voting paradox is equivalent to both the “coalition of minorities” and logrolling, it follows logically that these two are one and the same.

8 Riker and Brams, pp. 1236–40.

9 Ibid., p. 1241.

10 Except for minor notational differences this is taken directly from Table 5, Riker and Brams, p. 1241.

11 This concept is developed in some detail in Riker and Brams, p. 1239.

12 Riker and Brams, p. 1237.

13 The primary concern is not the actual occurrence of a particular outcome following a trade. The important point is that if either votes or intentions to vote are exchanged, there is no equilibrium.

14 This trade sounds somewhat strange since ordinarily one member does not promise another a vote on an issue which the latter has already won. What this does, of course, is enable m 2to break his previous agreement with m 3, who has promised to vote and be assured that still wins. The reason for using this device is that it allows the number of issues in the example to be kept to a minimum.

The question of the effect of broken agreements on continued vote trading is not a primary concern given the perspective of this paper. The matter is discussed by Mueller, Dennis C., “The Possibility of a Social Welfare Function: Comment,” American Economic Review, 57 (12, 1967), 1304–11Google Scholar, and by Coleman, James, “The Possibility of a Social Welfare Function: Reply,” American Economic Review, 57 (12, 1967), 1311–17Google Scholar in a response to R. E. Park, “Comment.”

15 The purpose of this argument is not to suggest that whenever votes are exchanged in the real world a frantic process of cycling commences. Rather it is to show that when votes can be exchanged, instability follows logically, whether or not it takes place in practice.

16 The preference configurations which lead to the cyclical majority are discussed in Riker, William H. and Ordeshook, Peter C., An Introduction to Positive Political Theory (Englewood Cliffs, New Jersey: Prentice-Hall, 1973), pp. 94–96 Google Scholar.

17 Riker and Ordeshook, pp. 100–109.

18 May, Kenneth O., “A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision,” Econometrica, 20 (10, 1952), 680–684 CrossRefGoogle Scholar.

19 Sen, Amartya K., “A Possibility Theorem on Majority Decisions,” Econometrica, 34 (04, 1966), 491–99CrossRefGoogle Scholar, has shown that the cyclical majority can be avoided by assuming Value-Restricted Preferences: “… a set of individual preferences over a triple of alternatives such that there exist one alternative and one value with the characteristic that the alternative never has that value in any individual' preference ordering …” This is the opposite of the condition stated above to insure cycling in the three member-three alternative case, that “… each outcome must be given a different rank by each member.”

20 Black, Duncan, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958) pp. 1–51 Google Scholar. The condition of single-peakedness assumes that all members' preferences over a set of motions can be ordered on a single dimension such that each preference function “… changes its direction at most once, from up to down.”

21 Plott, Charles R., “Rationality and Relevance in Social Choice Theory,” Social Science Working Paper No. 5 (California Institute of Technology, 08, 1971), pp. 10–13 Google Scholar. discusses Condition III and the example from Arrow, pp. 26–28.

22 Oppenheimer, , “Relating Coalitions,” pp. 5–6 Google Scholar.

23 Riker and Brams, p. 1242, show that the worst a member can do occurs when others trade and he refuses to join in.

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