Increasing Interdependence in Multivariate Distributions

This paper compares n-dimensional random vectors in terms of their interdependence. We adopt the stochastic dominance approach, relating orderings of interdependence expressed directly in terms of joint probability distributions to orderings 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 marginal distributions, attitudes towards interdependence must be represented through non-separability properties. We argue that the property of supermodularity (Topkis, 1978) of an objective function is a natural property with which to capture a preference for greater interdependence. Accordingly, we seek to characterize a partial ordering on joint distributions, with identical marginals, which is equivalent to one distribution’s yielding a higher expectation than another for all supermodular objective functions. Following the statistics literature, we refer to this partial ordering as the “supermodular stochastic ordering” (Shaked and Shanthikumar, 1997).