Now what if X believes itself to be well-calibrated only with probability 99%, in the following sense. Define S1 and S2 as before. What if, for all p, the probability X assigns to S2 is between p and 0.99p times the probability X assigns to S1?
This doesn’t handle well near 0, since it basically specifies a certain range on a log-scale (unless this was intentional?). You might want to weaken this to “for all p, the probability X assigns to S2 is between p+0.01 and p-0.01 times the probability X assigns to S1”.
I don’t follow. We are discussing agents that can prove that, for all S1, S2 as specified, S2 < 0.01+S1. This does not say how much less. It is possible that S1=S2, we just aren’t concerned with proving that in this thought experiment.
Slight nitpick:
This doesn’t handle well near 0, since it basically specifies a certain range on a log-scale (unless this was intentional?). You might want to weaken this to “for all p, the probability X assigns to S2 is between p+0.01 and p-0.01 times the probability X assigns to S1”.
You are quite right; I will change the post accordingly.
But that’s overdoing it. I’d invariably enter the lottery because it has an expected 0.5% chance of success.
I don’t follow. We are discussing agents that can prove that, for all S1, S2 as specified, S2 < 0.01+S1. This does not say how much less. It is possible that S1=S2, we just aren’t concerned with proving that in this thought experiment.