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.
Yes, you can predict that the outcomes of e.g. the coin flip will have 50-50 distribution instead of "the coin will land heads up" but this is a pretty useless kind of "prediction"; this trivial fixed point corresponds to "I don't know, and I won't even bother to try to know"
Also, consider the game of hunter-and-prey from my reply to the sibling comment. Again, the prediction that the prey will make some choice is arguably not a prediction at all.
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.