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On conditioning and consistency for nonlinear functionals

Author

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  • Edoardo Berton
  • Alessandro Doldi
  • Marco Maggis

Abstract

We consider a family of conditional nonlinear expectations defined on the space of bounded random variables and indexed by the class of all the sub-sigma-algebras of a given underlying sigma-algebra. We show that if this family satisfies a natural consistency property, then it collapses to a conditional certainty equivalent defined in terms of a state-dependent utility function. This result is obtained by embedding our problem in a decision theoretical framework and providing a new characterization of the Sure-Thing Principle. In particular we prove that this principle characterizes those preference relations which admit consistent backward conditional projections. We build our analysis on state-dependent preferences for a general state space as in Wakker and Zank (1999) and show that their numerical representation admits a continuous version of the state-dependent utility. In this way, we also answer positively to a conjecture posed in the aforementioned paper.

Suggested Citation

  • Edoardo Berton & Alessandro Doldi & Marco Maggis, 2024. "On conditioning and consistency for nonlinear functionals," Papers 2401.09054, arXiv.org, revised May 2024.
  • Handle: RePEc:arx:papers:2401.09054
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    References listed on IDEAS

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    1. Castagnoli, Erio & LiCalzi, Marco, 2006. "Benchmarking real-valued acts," Games and Economic Behavior, Elsevier, vol. 57(2), pages 236-253, November.
    2. Stanca, Lorenzo, 2020. "A simplified approach to subjective expected utility," Journal of Mathematical Economics, Elsevier, vol. 87(C), pages 151-160.
    3. Marco Frittelli & Marco Maggis, 2011. "Conditional Certainty Equivalent," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(01), pages 41-59.
    4. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, December.
    5. Peter P. Wakker & Horst Zank, 1999. "State Dependent Expected Utility for Savage's State Space," Mathematics of Operations Research, INFORMS, vol. 24(1), pages 8-34, February.
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