Multidimensional Risk Made Easy

2026-07-01

AI Summary

This study addresses the construction of certainty equivalents for multidimensional risks that satisfy three fundamental axioms: law invariance, vector stochastic dominance monotonicity, and invariance under independent background risks. By projecting multidimensional risks onto a scalar representation and leveraging the positive mixture representation of scalar entropy-based certainty equivalents, the paper provides the first complete characterization of multidimensional certainty equivalents fulfilling all three axioms. The proposed construction is shown to be equivalent to a robust preference ordering that remains invariant when independent background risks are introduced. Furthermore, within a social welfare framework, the shadow valuations derived from this approach correspond precisely to welfare weights, thereby offering a parsimonious and axiomatically grounded method for evaluating multidimensional risks.

Abstract

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.

Problem

• multidimensional risk
• certainty equivalent
• vector stochastic dominance
• background risk
• law-invariance

Innovation

• multidimensional risk
• certainty equivalent
• entropic preference
• stochastic dominance
• background risk