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Nonstandard utilities for lexicographically decomposable orderings

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  • Rizza, Davide

Abstract

Using a basic theorem from mathematical logic, I show that there are field-extensions of R on which a class of orderings that do not admit any real-valued utility functions can be represented by uncountably large families of utility functions. These are the lexicographically decomposable orderings studied in Beardon et al. (2002a). A corollary to this result yields an uncountably large family of very simple utility functions for the lexicographic ordering of the real Cartesian plane. I generalise these results to the lexicographic ordering of Rn, for every n>2, and to lexicographic products of lexicographically decomposable chains. I conclude by showing how almost all of these results may be obtained without any appeal to the Axiom of Choice.

Suggested Citation

  • Rizza, Davide, 2015. "Nonstandard utilities for lexicographically decomposable orderings," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 105-109.
    • Handle: RePEc:eee:mateco:v:60:y:2015:i:c:p:105-109
    DOI: 10.1016/j.jmateco.2015.06.012
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    References listed on IDEAS

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    1. Beardon, Alan F. & Candeal, Juan C. & Herden, Gerhard & Indurain, Esteban & Mehta, Ghanshyam B., 2002. "Lexicographic decomposition of chains and the concept of a planar chain," Journal of Mathematical Economics, Elsevier, vol. 37(2), pages 95-104, April.
    2. Fishburn, Peter C. & Lavalle, Irving H., 1991. "Nonstandard nontransitive utility on mixture sets," Mathematical Social Sciences, Elsevier, vol. 21(3), pages 233-244, June.
    3. Beardon, Alan F. & Candeal, Juan C. & Herden, Gerhard & Indurain, Esteban & Mehta, Ghanshyam B., 2002. "The non-existence of a utility function and the structure of non-representable preference relations," Journal of Mathematical Economics, Elsevier, vol. 37(1), pages 17-38, February.
    4. Lehmann, Daniel, 2001. "Expected Qualitative Utility Maximization," Games and Economic Behavior, Elsevier, vol. 35(1-2), pages 54-79, April.
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