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gjm comments on The Value of Uncertainty Reduction - references to academic literature appreciated - Less Wrong
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The equilibrium probability might not be well defined. (E.g., if for whatever reason you form a sufficiently firm intention to falsify whatever the oracle tells you.)
And yes, if the oracle tells you something about your own future actions -- which it has to, to give you an equilibrium probability -- it's unsurprising that you're going to feel a loss of freedom. Either that, or disbelieve the oracle.
I guess it would still be well-defined as a fixpoint though, like in "Closed Timelike Curves Make Quantum and Classical Computing Equivalent". Although by the same paper, it would be computationally infeasible for a predictor to actually find the fixpoint...
The point (or at least a point) is that there might not be a fixed point. I suppose what that might mean is that in a universe containing such a predictor you're unable to form such intentions as would lead to the failure. This seems sufficiently far removed from the real world that it would probably be better to consider scenarios that don't flirt so brazenly with paradox.
Yeah, you're are right.
The closed-timelike-curve paper weakens the guarantee of the predictor, so instead of saying "A with 25% probability, B with 75% probability", it will stochastically say either "A" or "B". So the kind of fixpoint it considers is less impressive.
What does it mean for a probability not to be well defined in this context? I mean, I think I share the intuition, but I'm not really comfortable with it either. Doesn't it seem strange that a probability could be well defined until I start learning more about it and trying to change it? How little do I have to care about the probability before it becomes well defined again?
As soon as the oracle is trying to make predictions that are affected by what the oracle says, the problem she has to solve shifts from "estimate the probabilities" to "choose what information to give, so as to produce consistent results given how that information will affect what happens". In some cases there might not be anything she can say that yields consistent results. Exactly where (if at all) that becomes impossible depends on the details of the situation.