This sort of Predictor is essentially computing a fixed point of a world function that takes the prediction as input and outputs a probability distribution over outcomes conditional on the Predictor making this prediction. If the world function is independent of the prediction, than this works normally, and there is one fixed point where you want it to be.

This raises the question of what the Predictor would do when the world function has multiple fixed points (e.g. U(p) = p), or no fixed points (e.g. U(p) = 1 if p<.5 else 0).