If your prediction is a probability distribution, rather than a discrete outcome label, then, assuming the distribution over the future depends continuously on your action, there should be a fixed point, I think?

Like, if you output a probability distribution among n options, and then there is a continuous map from the probability distribution you describe in your output, to another probability distribution over those options…

Err, hm, maybe you need a stronger hypothesis on the continuous map? If it is contractive then it will definitely have a fixed point. I don’t remember the hypothesis needed.