(U,V)-Ordering and a Duality Theorem for Risk Aversion and Lorenz-type Orderings

Abstract

There is a duality theory connecting certain stochastic orderings between cumulative distribution functions F_1,F_2 and stochastic orderings between their inverses F_1^(-1),F_2^(-1). This underlies some theories of utility in the case of the cdf and deprivation indices in the case of the inverse. Under certain conditions there is an equivalence between the two theories. An example is the equivalence between second order stochastic dominance and the Lorenz ordering. This duality is generalised to include the case where there is "distortion" of the cdf of the form v(F) and also of the inverse. A comprehensive duality theorem is presented in a form which includes the distortions and links the duality to the parallel theories of risk and deprivation indices. It is shown that some well-known examples are special cases of the results, including some from the Yaari social welfare theory and the theory of majorization.