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Extension of Monotonic Functions and Representation of Preferences

Author

Listed:
  • Özgür Evren

    (Department of Economics, New Economic School, Moscow 121353, Russian Federation)

  • Farhad Hüsseinov

    (School of Business, ADA University, AZ1008 Baku, Azerbaijan)

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.

Suggested Citation

  • Özgür Evren & Farhad Hüsseinov, 2021. "Extension of Monotonic Functions and Representation of Preferences," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1430-1451, November.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:4:p:1430-1451
    DOI: 10.1287/moor.2020.1095
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