The Supermodular Stochastic Ordering

Working Paper: CEPR ID: DP9486

Authors: Margaret A. Meyer; Bruno Strulovici

Abstract: In many economic applications involving comparisons of multivariate distributions, supermodularity of an objective function is a natural property for capturing a preference for greater interdependence. One multivariate distribution dominates another according to the `supermodular stochastic ordering' if it yields a higher expectation than the other for all supermodular objective functions. We prove that this ordering is equivalent to one distribution being derivable from another by a sequence of elementary, bivariate, interdependence-increasing transformations, and develop methods for determining whether such a sequence exists. For random vectors resulting from common and idiosyncratic shocks, we provide non-parametric sufficient conditions for supermodular dominance. Moreover, we characterize the orderings corresponding to supermodular objective functions that are also increasing or symmetric. We use the symmetric supermodular ordering to compare distributions generated by heterogeneous lotteries. Applications to welfare economics, committee decision-making, insurance, finance, and parameter estimation are discussed.

Keywords: Concordance; Copula; Correlation; Interdependence; Majorization; Mixture; Supermodular; Tournament

JEL Codes: D63; D81; G11; G22

Causal Claims

Cause	Effect
One multivariate distribution dominates another according to the supermodular stochastic ordering (C46)	It yields a higher expectation than the other for all supermodular objective functions (C79)
One distribution is derivable from another via elementary bivariate interdependence-increasing transformations (C46)	It increases the probability of joint high or low outcomes for pairs of variables while leaving the marginal distributions unchanged (C46)
Greater interdependence in random variables (C29)	Results in higher expected values under supermodular functions (C71)