Standard ROC curves look at how setting various thresholds on a continuous measure can be used to predict a two-level ordinal outcome (example: antibody level -> (not sick, sick) ). This can then be integrated over the whole range of thresholds yielding the classical Area Under the Curve (AUC).
The AUC has an interesting alternative definition: it is also the probability that, for a given pair of subjects, the continuous measure will rank them correctly with respect to the ordinal outcome.
In the case where this ordinal outcome has more than two levels (example: not sick, a little sick, very sick) , this alternative definition can be used to get a natural generalization of the concept of AUC.
My question is: does this generalization have a name? Is it a standard or at least known measure? Are there any more widely used alternatives?