Increasing interdependence of multivariate distributions

In many economic contexts, it is of interest to know whether one set of random variables displays a greater degree of interdependence than another. Orderings of interdependence are useful in the assessment of ex post inequality under uncertainty; in comparisons of multidimensional inequality; in assessments of the degree of conformity of behavior in social learning situations; in comparisons of the efficiency of matching institutions; in the valuation of portfolios of assets or insurance policies; and in assessments of systemic risk. This paper explores five orderings of interdependence for multivariate distributions: greater weak association, the supermodular ordering, the convex-modular ordering, the dispersion ordering, and the concordance ordering. We show that for two dimensions, all five orderings are equivalent, whereas for an arbitrary number of dimensions n > 2, the five orderings are strictly ranked. For the special case of binary random variables, we establish some equivalence results among the orderings. We conclude by illustrating the application of our orderings to the comparison of interdependence in behavior in a model of learning in networks and to the assessment of ex post inequality under uncertainty. ∗Email addresses: [email protected] [email protected]