We discuss equivalent axiomatic characterizations of distortion risk measures, and give a novel and concise proof of the characterization of elicitable distortion risk measures. Elicitability has recently been discussed as a desirable criterion for risk measures, motivated by statistical considerations of forecasting. We reveal the mathematical conflict between the requirements of elicitability and comonotonic additivity which intuitively explains why only Value-at-Risk and the mean are elicitable distortion risk measures in a general sense.
