A Critical Comparison of Three Notions of Fractional Stochastic Dominance

Abstract

Two notions of fractional stochastic dominance are recently proposed by Muller et al. (2017) and Huang et al. (2020), respectively. Our main objective is to understand the comparative advantages of the two notions, as well as their suitability in different contexts, by establishing several new technical results. For a more comprehensive comparison, we further include a third natural notion of fractional stochastic dominance based on the coefficient of relative risk aversion. Among the three notions, it turns out that one can be seen as a logarithmic version of second-order stochastic dominance (SSD), another one can be seen as a power version of SSD, whereas there does not exist a transformation to associate the last one with SSD. We find that these notions of fractional stochastic dominance are naturally connected to five classes of risk measures, including Value-at-Risk, Expected Shortfall, expectiles, entropic risk measures, and loss certainty equivalents. The three notions are further characterized in the contexts of the rank-dependent utility model and the cumulative prospect theory. We make some recommendations on which notion to use in specific situations, as they all have their own merits.