Searching for a Debreu’s Open Gap Lemma for Semiorders
| Autor: | Asier Estevan Muguerza |
|---|---|
| Přispěvatelé: | Universidad Pública de Navarra. Departamento de Estadística, Informática y Matemáticas, Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa. INAMAT2 - Institute for Advanced Materials and Mathematics, Nafarroako Unibertsitate Publikoa. Estatistika, Informatika eta Matematika Saila |
| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Mathematical Topics on Representations of Ordered Structures and Utility Theory ISBN: 9783030342258 Academica-e. Repositorio Institucional de la Universidad Pública de Navarra instname |
| Popis: | 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). The author acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under grants MTM2015-63608-P and ECO2015-65031. |
| Databáze: | OpenAIRE |
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