Searching for a Debreu’s open gap lemma for semiorders
Fecha
2020Autor
Versión
Acceso abierto / Sarbide irekia
Tipo
Capítulo de libro / Liburuen kapitulua
Versión
Versión aceptada / Onetsi den bertsioa
Impacto
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10.1007/978-3-030-34226-5_5
Resumen
In 1956 R. D. Luce introduced the notion of a semiorder to deal with indifference relations in the representation of a preference. During several years the problem of finding a utility function was studied until a representability characterization was found. However, there was almost no results on the continuity of the representation. A similar result to Debreu’s Lemma, but for semiorders was nev ...
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In 1956 R. D. Luce introduced the notion of a semiorder to deal with indifference relations in the representation of a preference. During several years the problem of finding a utility function was studied until a representability characterization was found. However, there was almost no results on the continuity of the representation. A similar result to Debreu’s Lemma, but for semiorders was never achieved. In the present paper we propose a characterization for the existence of a continuous representation (in the sense of Scott-Suppes) for bounded semiorders. As a matter of fact, the weaker but more manageable concept of ε-continuity is properly introduced for semiorders. As a consequence of this study, a version of the Debreu’s Open Gap Lemma is presented (but now for the case of semiorders) just as a conjecture, which would allow to remove the open-closed and closed-open gaps of a subset S ⊆ R, but now keeping the constant threshold, so that x + 1 < y if and only if g(x) + 1 < g(y) (x, y ∈ S). [--]
Materias
Semiorders,
Debreu’s Open Gap Lemma
Editor
Springer
Publicado en
Bosi, G., Campión, M., Candeal, J., Indurain, E. (eds.) Mathematical Topics on Representations of Ordered Structures and Utility Theory. Springer: Cham, 2020, pp. 109-128. ISBN 978-3-030-34225-8.
Departamento
Universidad Pública de Navarra. Departamento de Estadística, Informática y Matemáticas /
Nafarroako Unibertsitate Publikoa. Estatistika, Informatika eta Matematika Saila /
Universidad Pública de Navarra/Nafarroako Unibertsitate Publikoa. Institute for Advanced Materials and Mathematics - INAMAT2
Versión del editor
Entidades Financiadoras
The author acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under grants MTM2015-63608-P and ECO2015-65031.