Are Shortfall Systemic Risk Measures One Dimensional?

Shortfall systemic (multivariate) risk measures $\rho$ defined through an $N$-dimensional multivariate utility function $U$ and random allocations can be represented as classical (one dimensional) shortfall risk measures associated to an explicitly determined $1$-dimensional function constructed from $U$. This finding allows for simplifying the study of several properties of $\rho$, such as dual representations, law invariance and stability.