Interdependence in Multivariate Distributions: Stochastic Dominance Theorems and an Application to the Measurement of Ex Post Inequality under Uncertainty


Margaret A. Meyer

Nuffield College, Oxford University



This paper analyzes the measurement of interdependence among n random variables. We adopt the stochastic dominance approach, relating concepts of interdependence expressed directly in terms of joint probability distributions to concepts expressed indirectly through properties of objective functions whose
expectations are used to evaluate distributions. Since the expected values of additively separable objective functions depend only on the marginal distributions, attitudes towards correlation must be represented through non-separability properties. For the bivariate case, we present two stochastic dominance theorems, characterizing rankings of different strengths. For the multivariate case, we propose and characterize three different rankings, each a natural extension of one of the bivariate ones.

This analysis of interdependence is applicable to a wide range of problems in choice theory and welfare economics. Our results are presented in the context of one such application: the measurement of inequality in an uncertain environment. This context motivates the “tournament axiom”, which we view as a requirement for an objective function to represent a suitably strong aversion to negative interdependence. We apply our three multivariate stochastic dominance conditions to verify that the corresponding classes of objective functions satisfy the tournament axiom. We also analyze the relationships between our dominance conditions and the concepts of affiliation and association.