Continuous Representations of Preferences by Means of Two Continuous Functions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909104863182848 |
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| author | Bosi, Gianni Estevan, Asier |
| author_facet | Bosi, Gianni Estevan, Asier |
| contents | Let $\precsim$ be a reflexive binary relation on a topological space $(X, τ)$. A pair $(u,v)$ of continuous real-valued functions on $(X, τ)$ is said to be a {\em continuous representation} of $\precsim$ if, for all $x,y \in X$, [$(x \precsim y \Leftrightarrow u(x) \leq v(y))$]. In this paper we provide a characterization of the existence of a continuous representation of this kind in the general case when neither the functions $u$ and $v$ nor the topological space $(X,τ)$ are required to satisfy any particular assumptions. Such characterization is based on a suitable continuity assumption of the binary relation $\precsim$, called {\em weak continuity}. In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07908 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuous Representations of Preferences by Means of Two Continuous Functions Bosi, Gianni Estevan, Asier Theoretical Economics Let $\precsim$ be a reflexive binary relation on a topological space $(X, τ)$. A pair $(u,v)$ of continuous real-valued functions on $(X, τ)$ is said to be a {\em continuous representation} of $\precsim$ if, for all $x,y \in X$, [$(x \precsim y \Leftrightarrow u(x) \leq v(y))$]. In this paper we provide a characterization of the existence of a continuous representation of this kind in the general case when neither the functions $u$ and $v$ nor the topological space $(X,τ)$ are required to satisfy any particular assumptions. Such characterization is based on a suitable continuity assumption of the binary relation $\precsim$, called {\em weak continuity}. In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases. |
| title | Continuous Representations of Preferences by Means of Two Continuous Functions |
| topic | Theoretical Economics |
| url | https://arxiv.org/abs/2402.07908 |