Updating Utility Functions on Preordered Sets

Pavel Chebotarev ()
Additional contact information
Pavel Chebotarev: Technion–Israel Institute of Technology, Haifa 3200003, Israel

Mathematics, 2023, vol. 11, issue 22, 1-18

Abstract: We consider the problem of extending a function f P defined on a subset P of an arbitrary set X to X strictly monotonically with respect to a preorder ≽ defined on X , without imposing continuity constraints. We show that whenever ≽ has a utility representation, f P is extendable if and only if it is gap-safe increasing. This property means that whenever x ′ ≻ x , the infimum of f P on the upper contour of x ′ exceeds the supremum of f P on the lower contour of x , where x , x ′ ∈ X ˜ and X ˜ is X completed with two absolute ≽-extrema and, moreover, f P is weakly increasing. The completion of X makes the condition sufficient. The proposed method of extension is flexible in the sense that for any bounded utility representation u of ≽, it provides an extension of f P that coincides with u on a region of X that includes the set of P -neutral elements of X . An analysis of related topological theorems shows that the results obtained are not their consequences. The necessary and sufficient condition of extendability and the form of the extension are simplified when P is a Pareto set.

Keywords: extension of utility functions; monotonicity; utility representation of a preorder; lifting theorems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/11/22/4688/pdf (application/pdf)
https://www.mdpi.com/2227-7390/11/22/4688/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:11:y:2023:i:22:p:4688-:d:1282718

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().