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Collective Choice, Separation of Issues and Vote Trading*

Published online by Cambridge University Press:  01 August 2014

Thomas Schwartz
Thomas Schwartz*
Affiliation:
University of Texas at Austin
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Abstract

In legislatures and committees, a number of issues are voted on separately, leading to an outcome consisting of positions on each of these issues. I investigate the effects this separation of issues has on collective choices, assuming a very abstract collective choice model, whose assumptions are presupposed by many less abstract models, notably spatial models. Assuming the model, if there exists an undominated outcome (one to which no winning coalition prefers any other feasible outcome), it must be chosen in the absence of vote trading, although vote trading can (perversely) lead to a very different outcome. But vote trading does not necessarily lead to a “voting paradox” situation, contrary to several recent papers. The model enables us to define a natural solution concept for the case where every feasible outcome is dominated. Variations on this concept are explored. The effects of weakening the model are investigated.

Type
Articles
Information
American Political Science Review , Volume 71 , Issue 3 , September 1977 , pp. 999 - 1010
Copyright
Copyright © American Political Science Association 1977

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Footnotes

*

I presented earlier drafts of this paper at a Conference on the Foundations of Political Economy, University of Texas at Austin, February 1975, and at the annual meetings of the Public Choice Society, Chicago, April 1975. I profitted from the comments of the discussants, Peter Fishburn, Russel Hardin and Joe Oppenheimer at Austin, and Steven Brams at Chicago.

I am grateful to Joe Oppenheimer for stimulating discussions on the topic on various occasions, and to Martin Baily and Peter Bernholz for discovering a number of important errors. I owe a special debt of gratitude to Bernholz. This paper grew out of an extensive, extremely fruitful correspondence with Prof. Bernholz, occasioned by his paper, “Log-Rolling, Arrow-Paradox and Decision Rules,” Kyklos, 21 (November, 1974), 49–62.

References

1 Kadane, Joseph B., “On Division of the Question,” Public Choice, 13 (Fall, 1972), 47–54 CrossRefGoogle Scholar.

2 Oppenheimer, Joe A., “Relating Coalitions of Minorities to the Voter's Paradox, or Putting the Fly in the Democratic Pie,” paper delivered at the Annual Meeting of the South West Political Science Association, San Antonio, 03 30-April 1, 1972 Google Scholar.

3 Bernholz, Peter, “Log-Rolling, Arrow-Paradox and Cyclical Majorities,” Public Choice, 15 (Summer, 1973), 87–102 CrossRefGoogle Scholar; and Bernholz, “Log-Rolling, Arrow-Paradox and Decision Rules,” cited in acknowledgment note above.

4 More or less this result was proved, in a less general form, by Kadane, “On Division of the Question.”

5 On the early history of social choice theory, see Black, Duncan, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1963), Part IIGoogle Scholar.

6 Bernholz, “Log-Rolling, Arrow-Paradox and Cyclical Majorities,” and “Log-Rolling, Arrow-Paradox and Decision Rules.”

7 Oppenheimer, “Relating Coalitions of Minorities to the Voter's Paradox.”

8 Koehler, David H., “Vote-Trading and the Voting Paradox: A Proof of Logical Equivalence,” American Political Science Review, 69 (09, 1975), 954–960 CrossRefGoogle Scholar; and “Vote Trading and the Voting Paradox: Reply,” American Political Science Review, 69 (09, 1975)Google Scholar.

9 Bernholz, Peter, “Logrolling and the Paradox of Voting: Are They Really Logically Equivalent?” American Political Science Review, 69 (09, 1975), 961–962 CrossRefGoogle Scholar.

10 Oppenheimer, Joe A., “Some Political Implications of ‘Vote-Trading and the Voting Paradox: A Proof of Logical Equivalence,’” American Political Science Review, 69 (09, 1975), 963–966 CrossRefGoogle Scholar.

The thesis stated in this paper and in Bernholz, “Logrolling, Arrow-Paradox and Decision Rules,” is more general than that of Oppenheimer, “Relating Coalitions of Minorities to the Voter's Paradox,” and Bernholz, “Logrolling, Arrow Paradox and Cyclical Majorities.” Koehler in “Vote-Trading and the Voting Paradox: A Proof of Logical Equivalence,” restates the argument for the special three-voter example of Riker, William H. and Brams, Steven J., “The Paradox of Vote Trading,” American Political Science Review, 67 (12, 1973), 1235–1247 CrossRefGoogle Scholar, then makes a questionable attempt to generalize it.

Bernholz, “Logrolling and the Paradox of Voting: Are They Really Logically Equivalent?” points out that Koehler's general formulation is too general and that Koehler falsely assumes there are no P-cycles if there are no three-element P-cycles. Koehler, “Vote-Trading and the Voting Paradox: Reply” responds by citing a trivial fact of elementary logic: there are no P-intransitivities if there are no three-element P-intransitivities. True. But not every P-intransitivity is a P-cycle.

11 If nothing in Ω is stable, there must exist a P-cycle. But not conversely. A P-cycle can toexist with a stable outcome. Of course, if there is a P-rcycle, then, even if ft itself has a stable member, some subset of Ω must lack a stable member.

12 See Schwartz, Thomas, “On the Possibility of Rational Policy Evaluation,” Theory and Decision, 1 (01, 1970), 1–25 CrossRefGoogle Scholar; and “Rationality and the Myth of the Maximum,” Noȗs, 7 (05, 1972), 97–117 Google Scholar.

13 Schwartz, Thomas, “Serial Collective Choice,” paper presented at the annual meeting of the Public Choice Society, New Haven, 04 1974 Google Scholar.

14 Tullock, Gordon, “Problems of Majority Voting,” Journal of Political Economy, 67 (12, 1959), 571–579 CrossRefGoogle Scholar, reprinted in Freedom and Authority: An Introduction to Social and Political Philosophy, ed. Schwartz, Thomas (Encino, Ca.: Dickenson, 1973), pp. 295–303 Google Scholar.

15 Riker and Brams, “The Paradox of Vote Trading.”

16 Tullock's argument is criticized by Downs, Anthony, “In Defense of Majority Voting,” Journal of Political Economy, 69 (04, 1961), 192–199 CrossRefGoogle Scholar, reprinted in Freedom and Authority, ed. Schwartz, , pp. 304–313 Google Scholar, and defended against Downs's criticism by Tullock, Gordon, “Reply to a Traditionalist,” Journal of Political Economy, 69 (04, 1961), 200–203 CrossRefGoogle Scholar.

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