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Is the inf-convolution of law-invariant preferences law-invariant?

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  • Liu, Peng
  • Wang, Ruodu
  • Wei, Linxiao

Abstract

We analyze the question of whether the inf-convolution of law-invariant risk functionals (preferences) is still law-invariant. In other words, we try to understand whether the representative economic agent (after risk redistribution) only cares about the distribution of the total risk, assuming all individual agents do so. Although the answer to the above question seems to be affirmative for many examples of commonly used risk functionals in the literature, the situation becomes delicate without assuming specific forms and properties of the individual functionals. We illustrate with examples the surprising fact that the answer to the main question is generally negative, even in an atomless probability space. Furthermore, we establish a few very weak conditions under which the answer becomes positive. These conditions do not require any specific forms or convexity of the risk functionals, and they are the richness of the underlying probability space, and monotonicity or continuity of one of the risk functionals. We provide several examples and counter-examples to discuss the subtlety of the question on law-invariance.

Suggested Citation

  • Liu, Peng & Wang, Ruodu & Wei, Linxiao, 2020. "Is the inf-convolution of law-invariant preferences law-invariant?," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 144-154.
  • Handle: RePEc:eee:insuma:v:91:y:2020:i:c:p:144-154
    DOI: 10.1016/j.insmatheco.2020.01.004
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    Cited by:

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    3. Marcelo Brutti Righi & Marlon Ruoso Moresco, 2020. "Inf-convolution and optimal risk sharing with countable sets of risk measures," Papers 2003.05797, arXiv.org, revised Mar 2022.
    4. Shuo Gong & Yijun Hu & Linxiao Wei, 2022. "Distortion risk measures in random environments: construction and axiomatic characterization," Papers 2211.00520, arXiv.org, revised Mar 2023.
    5. Shengzhong Chen & Niushan Gao & Denny Leung & Lei Li, 2021. "Automatic Fatou Property of Law-invariant Risk Measures," Papers 2107.08109, arXiv.org, revised Jan 2022.
    6. Erio Castagnoli & Giacomo Cattelan & Fabio Maccheroni & Claudio Tebaldi & Ruodu Wang, 2021. "Star-shaped Risk Measures," Papers 2103.15790, arXiv.org, revised Apr 2022.
    7. Felix-Benedikt Liebrich, 2021. "Risk sharing under heterogeneous beliefs without convexity," Papers 2108.05791, arXiv.org, revised May 2022.
    8. Jean-Gabriel Lauzier & Liyuan Lin & Ruodu Wang, 2024. "Negatively dependent optimal risk sharing," Papers 2401.03328, arXiv.org.

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