# Social choice theory and citizens' intransitive weak preference—A paradox

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Public Finance Social Choice Choice Theory Social Choice Theory Weak Preference
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## References

- 1.Alfred Tarski, “Truth and Proof,”
*Scientific American*, June, 1969.Google Scholar - 2.K.O. May, “Intransitivity, Utility, and the Aggregation of Preference Patterns,”
*Econometrica*, XXII, 1954.Google Scholar - 2a.Thomas Schwartz, “On the Possibility of Rational Policy Evaluation,”
*Theory and Decision*, 1, 1970.Google Scholar - 3.
*Ibid.*Google Scholar - 4.A.K. Sen,
*Collective Choice and Social Welfare*, Holden Day, 1970, pp. 10, 15, 16, 81.Google Scholar - 4a.J. von Neumann and O. Morgenstern,
*The Theory of Games and Economic Behavior*, Princeton, 1947, pp. 39–40.Google Scholar - 5.R. Duncan Luce, “Semiorders and a Theory of Utility Discrimination,”
*Econometrica*, Vol. 24, 1956, pp. 178–191. Luce characterizes a class of relations called semiorders from which he proves an interesting representation theorem. A version of his definition and theorem is as follows: Definition: ≥is a semiorder on A iff ≥is a binary relation on A and for every x,y,z and w in A, 1) Either x≥y or y≥x; 2) If not y≥x, and y≥z and z≥y and not w≥z then not w≥x; 3) If neither y≥x nor z≥y, and y≥w and w≥y, then not both (x≥w and w≥x) and (z≥w and w≥z). Theorem 1: If u is a function from A into the reals and ε is a positive real and ≥is a relation on A and for every s,y in A, x≥y iff, u(x)>u (y)+ε or |u(x)−u(y)|≤ε, then ≥is a semiorder on A. Theorem 2: If ≥is a semiorder on A and A is finite, then for some function u from A into the reals and some positive real ε, for every x, y in A, x≥y iff, u(x)>u(y)+ε or |u(x)−u(y)|≤ε.Google Scholar - 6.It should be noted that Assumption 1 is much weaker than the plausable assumption that the domain of C include all functions from I into semi-orders on S. This semiorder assumption would in turn, however, be much weaker than (implied by but does not imply) Fishburn's assumption, which the author has become aware of, that the domain of C includes all functions from I into strict partial orders on S. See: P.C. Fishburn,
*The Theory of Social Choice*, Princeton University Press, 1973, pp. 83–84.Google Scholar - 7.Dennis Packard, “Rational Ranking Functions for Cyclical Comparatives,” Dissertation, Stanford, 1974.Google Scholar
- 7a.Thomas Schwartz, “Rationality and the Myth of the Maximum,”
*Nous*, May, 1972.Google Scholar

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© Center for Study of Public Choice Virginia Polytechnic Institute and State University 1974