Extension of Monotonic Functions and Representation of Preferences

Özgür Evren () and Farhad Hüsseinov ()
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Özgür Evren: Department of Economics, New Economic School, Moscow 121353, Russian Federation
Farhad Hüsseinov: School of Business, ADA University, AZ1008 Baku, Azerbaijan

Mathematics of Operations Research, 2021, vol. 46, issue 4, 1430-1451

Abstract: Consider a dominance relation (a preorder) ≿ on a topological space X , such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set K ⊆ X , we study when a continuous real function on K that is strictly monotonic with respect to ≿ can be extended to X without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when X is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space X starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.

Keywords: Primary: 91B16; 54C05; 54C20; 54F05; secondary: 91B06; 91B14; 91B42; Primary: Mathematics: functions; utility/preference: choice functions; theory; preordered space; monotonicity; continuity; extension; dominance relation; revealed preference; rational choice; (multi) utility representation (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:46:y:2021:i:4:p:1430-1451

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