Dominance Relations Among Standardized Variables

This paper examines stochastic dominance relations among discrete random variables defined on a common integer domain. While these restrictions are minimal, they lead both to new theoretical results and to simpler proofs of existing one. The new results, obtained for dominance criteria of any degree, generalize an SSD result of Rothschild-Stiglitz to describe how for any dominance criterion a dominated variable is equal in distribution to a dominated variable plus perturbation terms. If the variables are comparable under FSD the perturbations are downward shift terms, while under SSD (TSD) all but two (three) of the perturbations are zero mean disturbance terms (noise). Under SSD the remaining perturbations are shift terms and under TSD noise and shift terms. However, under either SSD or TSD these remaining terms are identically zero if the variables to be compared have equal means. The paper also finds new proofs of well known results relating dominance criteria to preferences.