Increasing interdependence of multivariate distributions

Orderings of interdependence are useful in many economic contexts: in assessing ex post inequality under uncertainty; in comparing multidimensional inequality; in valuing portfolios of assets or insurance policies; and in assessing systemic risk. We explore five orderings of interdependence for multivariate distributions: greater weak association, the supermodular ordering, the convex-modular ordering, the dispersion ordering, and the concordance ordering. For two dimensions, all five are equivalent, whereas for three dimensions, the first four are strictly ranked and the last two are equivalent, and for four or more dimensions, all five are strictly ranked. For the special case of binary random variables, we establish some equivalences among the orderings.