Topologies for semicontinuous Richter–Peleg multi-utilities

Gianni Bosi, Asier Estevan () and Armajac Raventós-Pujol
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Asier Estevan: Public University of Navarre
Armajac Raventós-Pujol: Public University of Navarre
Authors registered in the RePEc Author Service: Armajac Raventós Pujol

Theory and Decision, 2020, vol. 88, issue 3, No 6, 457-470

Abstract: Abstract The present paper gives a topological solution to representability problems related to multi-utility, in the field of Decision Theory. Necessary and sufficient topologies for the existence of a semicontinuous and finite Richter–Peleg multi-utility for a preorder are studied. It is well known that, given a preorder on a topological space, if there is a lower (upper) semicontinuous Richter–Peleg multi-utility, then the topology of the space must be finer than the Upper (resp. Lower) topology. However, this condition fails to be sufficient. Instead of search for properties that must be satisfied by the preorder, we study finer topologies which are necessary or/and sufficient for the existence of semicontinuous representations. We prove that Scott topology must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter–Peleg multi-utility. However, the existence of this representation cannot be guaranteed. A sufficient condition is given by means of Alexandroff’s topology, for that, we prove that more order implies less Alexandroff’s topology, as well as the converse. Finally, the paper is implemented with a topological study of the maximal elements.

Keywords: Preorders; Multi-utility theory; Richter–Peleg; Semicontinuity (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s11238-019-09730-7

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