  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.
