Multidimensional Risk Made Easy

Suppose we want to assign a certainty equivalent--one number--to a multivariate risk.

Which such assignments are law-invariant, monotone with respect to vector stochastic dominance, and invariant to independent background risk?

I show that every such certainty equivalent is a positive mixture of scalar entropic certainty equivalents applied to positive projections of the vector risk.

The same representation yields a robust-order characterization: unanimity across such certainty equivalents is equivalent, up to closure, to dominance after adding independent multidimensional background risk.

In a social-welfare specialization, the corresponding shadow valuations are welfare weights.