Let’s imagine that we had a second order probability \(\mu\) defined over the set of (first order) probabilities \(P\). We could then reduce uncertainty to a single function by \(p^*(X) = \sum_P \mu(p)p(X)\) (if \(P\) is finite, in the interests of keeping things simple I discuss only this case). Now if \(p^*(X)\) is what is used in decision making, then there is no real sense in which we have a genuine IP model, and it cannot rationalise the Ellsberg choice, nor can it give rise to incomparability. If there is some alternative use that \(\mu\) is put to, a use that allows incomparability and that rationalises Ellsberg choices, then it might be a genuine rival to credal sets, but it represents just as much of a departure from the orthodox theory as IP does.