It seems to me that the only relevant difference here is the set of agents that you perceive yourself as being correlated with, and that the update being from empirical observation vs. logical reasoning is unimportant.
Replace “Y1” and “Y2″ with empirical statements which are probabilistically independent according to my prior, say, “the Hubble constant is x” and “the physical baryon density is y” and replace “prove a statement” with “conduct a definitive cosmological experiment”, and everything seems to go through the same. (If the thought is that we want all agents in a large world to verify the same statements, that’s dealt with by having the empirical statements be about global cosmological parameters.)
As noted in this comment, if you took yourself to control both agents that saw “X is true” and “X is false” in the original Calculator Bet, then you bet at 99:1 odds. And in your example, if you took yourself to only control ‘agents who proved Y1’ (rather than ‘agents who proved a statement’), then you bet at 1:1 odds.
It seems to me that the only relevant difference here is the set of agents that you perceive yourself as being correlated with, and that the update being from empirical observation vs. logical reasoning is unimportant.
Replace “Y1” and “Y2″ with empirical statements which are probabilistically independent according to my prior, say, “the Hubble constant is x” and “the physical baryon density is y” and replace “prove a statement” with “conduct a definitive cosmological experiment”, and everything seems to go through the same. (If the thought is that we want all agents in a large world to verify the same statements, that’s dealt with by having the empirical statements be about global cosmological parameters.)
As noted in this comment, if you took yourself to control both agents that saw “X is true” and “X is false” in the original Calculator Bet, then you bet at 99:1 odds. And in your example, if you took yourself to only control ‘agents who proved Y1’ (rather than ‘agents who proved a statement’), then you bet at 1:1 odds.