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Continuous Multi-utility Representations of Preorders and the Chipman Approach
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: Received: 29 January 2024 / Approved: 30 January 2024 / Online: 30 January 2024 (04:01:45 CET)
A peer-reviewed article of this Preprint also exists.
Bosi, G.; Daris, R.; Zuanon, M. Continuous Multi-Utility Representations of Preorders and the Chipman Approach. Axioms 2024, 13, 148. Bosi, G.; Daris, R.; Zuanon, M. Continuous Multi-Utility Representations of Preorders and the Chipman Approach. Axioms 2024, 13, 148.
Abstract
Chipman contends, in stark contrast to the conventional view, that utility is not a real number but a vector and is inherently lexicographic in nature. According to these views, it will be proved that, for every preordered topological space (X,≾,t), the assumption t to be finer than the order topology t≾ on (X,t), i.e. t⊇t≾, and the assumption the quotient topology t∣∼≾ that is defined on the preordered set (X∣∼,≾∣∼) of indifference classes of ≾ to be Hausdorff imply that a cardinal number κ and a (complete) preorder ≲ on {0,1}κ that is coarser than the lexicographical ordering ≤lex on {0,1}κ, i.e. ≤lex⊂≲, can be chosen in such a way that there exists a continuous order-embedding ϑ:(X,≾,t)⟶({0,1}κ,≲,t≤lex). This theorem will be compared with a theorem that, in particular, describes necessary and sufficient conditions for ≾ to have a continuous multi-utility representation.
Keywords
Hausdorff space; Continuous multi-utility representation; order-embedding; semi-closed preorder
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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