Multi-fractional Stochastic Dominance : Mathematical Foundations

In the landmark article [MSTW17], Müller et. al. introduced the notion of fractional stochastic dominance (SD) to interpolate between first and second SD relations. In this article, we introduce a novel family of multi-fractional stochastic orders that generalizes fractional SD in a natural manner. The family of multi-fractional SD is parametrized by an arbitrary non-decreasing function γ ranging between 0 and 1 which provides the feature of local interpolation rather than a global one. We show that the multi-fractional (1 + γ)-SD is generated by a class of increasing utility functions allowing local non-concavity where the steepness of the non-concavity depends on its location and it is controlled by function γ. We also introduced the notion of local greediness that allows us, among other things, to systematically study multi-fractional utility class. The multi-fractional utility class is well-suited for representing a decision maker’s preferences in terms of risk aversion and greediness at a local level. Several basic properties as well as illustrating examplesare presented