Dual theory of choice with multivariate risks

TL;DR

This paper extends Yaari's dual theory of choice under risk to multivariate prospects, using optimal transportation to define multivariate quantiles and comonotonicity, and characterizes risk-averse decision makers.

Contribution

It introduces a multivariate dual theory framework using optimal transportation, extending univariate concepts to multidimensional risks and characterizing risk aversion.

Findings

Decision makers evaluate prospects via weighted sums of multivariate quantiles.

Multivariate quantiles and comonotonicity are defined through optimal transportation maps.

Risk aversion is characterized with local utility functions within this framework.

Abstract

We propose a multivariate extension of Yaari's dual theory of choice under risk. We show that a decision maker with a preference relation on multidimensional prospects that preserves first order stochastic dominance and satisfies comonotonic independence behaves as if evaluating prospects using a weighted sum of quantiles. Both the notions of quantiles and of comonotonicity are extended to the multivariate framework using optimal transportation maps. Finally, risk averse decision makers are characterized within this framework and their local utility functions are derived. Applications to the measurement of multi-attribute inequality are also discussed.