A note on relative efficiency and risk aversion

Stochastic dominance rules have recently been suggested as criteria for efficient choice under uncertainty. However, the aspect of relative efficiency has been neglected, i.e., given any specific utility function u₀, find the conditions imposed on F and G such that $\text{E}{}_{\mathrm{F}}\text{u≧E}{}_{\mathrm{G}}\text{u}$ for all u?U (u₀), where U(u₀)={u| ? u′/u′ ≥ ? u″₀/u′₀}. This note suggests a necessary and sufficient condition for relative efficiency in the case that F and G cross other only once. Some applications are discussed.