Characterizing Fractional Degree Stochastic Dominance by Invariance Laws

This paper investigates whether and when fractional degree stochastic dominance rules can inherit the invariance properties of integer-degree stochastic dominance. Within a general formulation, we first show that it is impossible for fractional degree stochastic dominance to obey all the six invariance laws enjoyed by integer-degree stochastic dominance, and then we identify a unique dominance rule for each subarray of mutually compatible invariance laws. We explore the invariance conditions ensuring that the "risk apportionment'' process is extendable to fractional degrees. This study facilitates the understanding of the differences among various prior rules, and provide a clear guideline for choosing an appropriate rule in applications, as illustrated by the derivation of an option pricing bound and two comparative statics exercises.