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Preferences Over Rich Sets of Random Variables: Semicontinuity in Measure versus Convexity


  • Alexander Zimper

    () (Department of Economics; University of Pretoria; postal address: Private Bag X20; Hat.eld 0028; South Africa)

  • Hirbod Assa

    () (Institute for Financial and Actuarial Mathematics and Institute for Risk and Uncertainty, University of Liverpool, Center for Doctoral Training, Chadwick Building, G62, Liverpool UK.)


The choice of a continuity concept in decision theoretic models has behavioral meaning because it pins down how the decision maker perceives the similarity of random variables. This paper analyzes the preferences of a decision maker who perceives similarity in accordance with the topology of convergence in measure. As our main insight we show that this decision maker cannot be globally risk or ambiguity averse whenever her preferences are lower-semicontinuous and complete on a rich set of random variables. Real life decision makers who perceive the similarity of random variables in accordance with convergence in measure might thus account for violations of global convexity as observed in empirical studies. Similarly, the non-convex risk measure value-at-risk might be popular among decision makers because it represents preferences that are lower-semicontinuous in measure.

Suggested Citation

  • Alexander Zimper & Hirbod Assa, 2019. "Preferences Over Rich Sets of Random Variables: Semicontinuity in Measure versus Convexity," Working Papers 201940, University of Pretoria, Department of Economics.
  • Handle: RePEc:pre:wpaper:201940

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    References listed on IDEAS

    1. Dekel, Eddie, 1989. "Asset Demands without the Independence Axiom," Econometrica, Econometric Society, vol. 57(1), pages 163-169, January.
    2. Alain Chateauneuf & Michéle Cohen & Isaac Meilijson, 2005. "More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 25(3), pages 649-667, April.
    3. Freddy Delbaen, 2009. "Risk Measures For Non‐Integrable Random Variables," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 329-333, April.
    4. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    5. Gilboa, Itzhak, 1987. "Expected utility with purely subjective non-additive probabilities," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 65-88, February.
    6. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    7. Gilboa, Itzhak, 1987. "Expected utility with purely subjective non-additive probabilities," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 65-88, February.
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    More about this item


    Similarity Perceptions; Continuous Preferences; Uncertainty; Ambiguity; Utility Representations; Risk Measures;

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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