Is the Inf-convolution of Law-invariant Preferences Law-invariant?

22 Pages Posted: 13 May 2019

See all articles by Peng Liu

Peng Liu

University of Waterloo - Department of Statistics and Actuarial Science

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Linxiao Wei

Wuhan University of Technology

Date Written: April 13, 2019

Abstract

We analyze the question of whether the inf-convolution of law-invariant risk functionals (preferences) is still law-invariant. In economic terms, this question means if all agents in a risk sharing system only care about the distributions of risks, whether the resulting (after risk redistribution) representative agent also only cares about the distribution of the total risk, regardless of how the total risk is defined as a random variable. We first illustrate with some examples that such an assertion is generally false. 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.

Keywords: law-invariance, inf-convolution, preferences, risk functionals, risk sharing

JEL Classification: C61, G10

Suggested Citation

Liu, Peng and Wang, Ruodu and Wei, Linxiao, Is the Inf-convolution of Law-invariant Preferences Law-invariant? (April 13, 2019). Available at SSRN: https://ssrn.com/abstract=3371642 or http://dx.doi.org/10.2139/ssrn.3371642

Peng Liu

University of Waterloo - Department of Statistics and Actuarial Science ( email )

200 University Avenue West
Waterloo, Ontario N2L 3G1
Canada

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

Linxiao Wei (Contact Author)

Wuhan University of Technology ( email )

Wuhan
China

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