  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.
