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Aggregation of Semiorders: Intransitive Indifference Makes a Difference

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  • Itzhak Gilboa
  • Robert Lapson

Abstract

A semiorder can be thought of as a binary relation P for which there is a utility "u" representing it in the following sense: xPy iff u(x)-u(y) > 1. We argue that weak orders (for which indifference is transitive) can not be considered a successful approximation of semiorders; for instance, a utility function representing a semiorder in the manner mentioned above is almost unique, i.e. cardinal and not only ordinal. In this paper we deal with semiorders on a product space and their relation to given semiorders on the original spaces. Following the intuition of Rubinstein we find surprising results: with the appropriate framework, it turns out that a Savage-type expected utility requires significantly weaker axioms than it does in the context of weak orders.

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Paper provided by Northwestern University, Center for Mathematical Studies in Economics and Management Science in its series Discussion Papers with number 870.

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Date of creation: Feb 1990
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Handle: RePEc:nwu:cmsems:870

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  1. Chateauneuf, Alain, 1987. "Continuous representation of a preference relation on a connected topological space," Journal of Mathematical Economics, Elsevier, vol. 16(2), pages 139-146, April.
  2. Ng, Yew-Kwang, 1975. "Bentham or Bergson? Finite Sensibility, Utility Functions and Social Welfare Functions," Review of Economic Studies, Wiley Blackwell, vol. 42(4), pages 545-69, October.
  3. Gensemer, Susan H., 1987. "Continuous semiorder representations," Journal of Mathematical Economics, Elsevier, vol. 16(3), pages 275-289, June.
  4. Gilboa, Itzhak, 1987. "Expected utility with purely subjective non-additive probabilities," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 65-88, February.
  5. Amos Tversky & Daniel Kahneman, 1979. "Prospect Theory: An Analysis of Decision under Risk," Levine's Working Paper Archive 7656, David K. Levine.
  6. Quiggin, John, 1982. "A theory of anticipated utility," Journal of Economic Behavior & Organization, Elsevier, vol. 3(4), pages 323-343, December.
  7. Avraham Beja & Itzhak Gilboa, 1989. "Numerical Representations of Imperfectly Ordered Preferences (A Unified Geometric Exposition," Discussion Papers 836, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  8. Rubinstein, Ariel, 1988. "Similarity and decision-making under risk (is there a utility theory resolution to the Allais paradox?)," Journal of Economic Theory, Elsevier, vol. 46(1), pages 145-153, October.
  9. John C. Harsanyi, 1955. "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility," Journal of Political Economy, University of Chicago Press, vol. 63, pages 309.
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Cited by:
  1. Manzini, Paola & Mariotti, Marco, 2004. "A Vague Theory of Choice over Time," IZA Discussion Papers 1228, Institute for the Study of Labor (IZA).
  2. Vila, Xavier, 1998. "On the Intransitivity of Preferences Consistent with Similarity Relations," Journal of Economic Theory, Elsevier, vol. 79(2), pages 281-287, April.
  3. Bouyssou, Denis & Pirlot, Marc, 2005. "Following the traces:: An introduction to conjoint measurement without transitivity and additivity," European Journal of Operational Research, Elsevier, vol. 163(2), pages 287-337, June.

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