Theory and Decision

, Volume 3, Issue 1, pp 18–40 | Cite as

Even-chance lotteries in social choice theory

  • Peter C. Fishburn
Article

Abstract

This paper discusses aspects of the theory of social choice when a nonempty choice set is to be determined for each situation, which consists of a feasible set of alternatives and a preference order for each voter on the set of nonempty subsets of alternatives. The individual preference assumptions include ordering properties and averaging conditions, the latter of which are motivated by the interpretation that subset A is preferred to subset B if and only if the individual prefers an even-chance lottery over the basic alternatives in A to an even-chance lottery over the basic alternatives in B. Corresponding to this interpretation, a choice set with two or more alternatives is resolved by an even-chance lottery over these alternatives. Thus, from the traditional no-lottery social choice theory viewpoint, ties are resolved by even-chance lotteries on the tied alternatives. Compared to the approach which allows all lotteries to compete along with the basic alternatives, the present approach is a contraction which allows only even-chance lotteries.

After discussing individual preference axioms, the paper examines Pareto optimality for nonempty subsets of a feasible set in a social choice context with n voters. Aspects of simple-majority comparisons in the even-chance context follow, including an analysis of single-peaked preferences. The paper concludes with an Arrowian type impossibility theorem that is designed for the even-chance setting.

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Bibliography

  1. E. W. Adams, ‘Elements of a Theory of Inexact Measurement’, Philosophy of Science 32 (1965) 205–228.Google Scholar
  2. K. J. Arrow, ‘A Difficulty in the Concept of Social Welfare’, Journal of Political Economy 58 (1950) 328–346.Google Scholar
  3. K. J. Arrow, ‘Rational Choice Functions and Orderings’, Economica 26 (1959) 121–127.Google Scholar
  4. K. J. Arrow, Social Choice and Individual Values, 2nd ed., New York 1963.Google Scholar
  5. D. Black, The Theory of Committees and Elections, Cambridge 1958.Google Scholar
  6. P. C. Fishburn, Utility Theory for Decision Making, New York 1970a.Google Scholar
  7. P. C. Fishburn, ‘Intransitive Indifference in Preference Theory: A Survey’, Operations Research 18 (1970b) 207–228.Google Scholar
  8. P. C. Fishburn, ‘A Location Theorem for Single-Peaked Preferences’, Journal of Economic Theory 4 (1972a) 94–97.Google Scholar
  9. P. C. Fishburn, ‘Lotteries and Social Choices’, Journal of Economic Theory 5 (1972b) in press.Google Scholar
  10. P. C. Fishburn, ‘Transitive Binary Social Choices and Intraprofile Conditions’, Econometrica (1973a) in press.Google Scholar
  11. P. C. Fishburn, ‘Social Choice Functions’, mimeographed (1973b).Google Scholar
  12. P. C. Fishburn, ‘Subset Choice Conditions and the Computation of Social Choice Sets’, mmimeographed (1973c).Google Scholar
  13. K. Inada, ‘Majority Rule and Rationality’, Journal of Economic Theory 2 (1970) 27–40.Google Scholar
  14. A. Mas-Colell, and H. Sonnenschein, ‘General Possibility Theorems for Group Decisions’, Review of Economic Studies 39 (1972) 185–192.Google Scholar
  15. P. K. Pattanaik, ‘On Social Choice with Quasitransitive Individual Preferences’, Journal of Economic Theory 2 (1970) 267–275.Google Scholar
  16. D. Scott, ‘Measurement Structures and Linear Inequalities’, Journal of Mathematical Psychology 1 (1964) 233–247.Google Scholar
  17. A. K. Sen, and P. K. Pattanaik, ‘Necessary and Sufficient Conditions for Rational Choice under Majority Decision’, Journal of Economic Theory 1 (1969) 178–202.Google Scholar
  18. K. A. Shepsle, ‘A Note on Zeckhauser's ‘Majority Rule with Lotteries on Alternatives’: The Case of The Paradox of Voting’, Quarterly Journal of Economics 84 (1970) 705–709.Google Scholar
  19. R. Zeckhauser, ‘Majority Rule with Lotteries on Alternatives’, Quarterly Journal of Economics 83 (1969) 696–703.Google Scholar

Copyright information

© D. Reidel Publishing Company 1972

Authors and Affiliations

  • Peter C. Fishburn
    • 1
  1. 1.College of Business Administration, The Pennsylvania State UniversityUniversity ParkUSA

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