+1 and many thanks for wading into this with me… I’ve been working all day and I’m still at work so can’t necessarily respond in full...

I agree that these problems are a lot simpler if reducing my uncertainty about X cannot help me affect X. This is not a minor class of problems. I’d love to have better information for a lot of problems in this class. That said, many of the problems that it seems most worthwhile for me to spend my time and money reducing my uncertainty about are of the type where I have a non-trivial role in how they play out. Assuming I do have some causal power over X, I think I’d pay a lot more to know the “equilibrium” probability of X after I’ve digested the information the oracle gave me—anything else seems like stale information… but learning that equilibrium probability seems weird as well. If I’m surprised by what the oracle says, then I imagine I’d ask myself questions like: how am I likely to react in regard to this information… what was the probability before I knew this information such that the current probability is what it is… It feels like I’m losing freedom… to what extent is the experience of uncertainty tied to the experience of freedom?

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?