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Debreu's open gap lemma for semiorders

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  • A. Estevan

Abstract

The problem of finding a (continuous) utility function for a semiorder has been studied since in 1956 R.D. Luce introduced in \emph{Econometrica} the notion. There was almost no results on the continuity of the representation. A similar result to Debreu's Lemma, but for semiorders, was never achieved. Recently, some necessary conditions for the existence of a continuous representation as well as some conjectures were presented by A. Estevan. In the present paper we prove these conjectures, achieving the desired version of Debreu's Open Gap Lemma for bounded semiorders. This result allows to remove the open-closed and closed-open gaps of a subset $S\subseteq \mathbb{R}$, but now keeping the constant threshold, so that $x+1

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  • A. Estevan, 2020. "Debreu's open gap lemma for semiorders," Papers 2010.04265, arXiv.org.
    • Handle: RePEc:arx:papers:2010.04265
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    1. Candeal, Juan Carlos & Indurain, Esteban & Zudaire, Margarita, 2002. "Numerical representability of semiorders," Mathematical Social Sciences, Elsevier, vol. 43(1), pages 61-77, January.
    2. Vincke, Philippe, 1980. "Linear Utility Functions on Semiordered Mixture Spaces," Econometrica, Econometric Society, vol. 48(3), pages 771-775, April.
    3. Fuad Aleskerov & Denis Bouyssou & Bernard Monjardet, 2007. "Utility Maximization, Choice and Preference," Springer Books, Springer, edition 0, number 978-3-540-34183-3, June.
    4. Gensemer, Susan H., 1987. "Continuous semiorder representations," Journal of Mathematical Economics, Elsevier, vol. 16(3), pages 275-289, June.
    5. Peter C. Fishburn, 1970. "Intransitive Indifference in Preference Theory: A Survey," Operations Research, INFORMS, vol. 18(2), pages 207-228, April.
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