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

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