Here's a puzzle I've been trying to figure out. It involves observation selection effects and agreeing to disagree. It is related to a paper I am writing, so help would be appreciated. The puzzle is also interesting in itself.

Charlie tosses a fair coin to determine how to stock a pond. If heads, it gets 3/4 big fish and 1/4 small fish. If tails, the other way around. After Charlie does this, he calls Al into his office. He tells him, "Infinitely many scientists are curious about the proportion of fish in this pond. They are all good Bayesians with the same prior. They are going to randomly sample 100 fish (with replacement) each and record how many of them are big and how many are small. Since so many will sample the pond, we can be sure that for any n between 0 and 100, some scientist will observe that n of his 100 fish were big. I'm going to take the first one that sees 25 big and team him up with you, so you can compare notes." (I don't think it matters much whether infinitely many scientists do this or just 3^^^3.)

Okay. So Al goes and does his sample. He pulls out 75 big fish and becomes nearly certain that 3/4 of the fish are big. Afterwards, a guy named Bob comes to him and tells him he was sent by Charlie. Bob says he randomly sampled 100 fish, 25 of which were big. They exchange ALL of their information.

Question: How confident should each of them be that 3/4 of the fish are big?

Natural answer: Charlie should remain nearly certain that ¾ of the fish are big. He knew in advance that someone like Bob was certain to talk to him regardless of what proportion of fish were big. So he shouldn't be the least bit impressed after talking to Bob.

But what about Bob? What should he think? At first glance, you might think he should be 50/50, since 50% of the fish he knows about have been big and his access to Al's observations wasn't subject to a selection effect. But that can't be right, because then he would just be agreeing to disagree with Al! (This would be especially puzzling, since they have ALL the same information, having shared everything.) So maybe Bob should just agree with Al: he should be nearly certain that ¾ of the fish are big.

But that's a bit odd. It isn't terribly clear why Bob should discount all of his observations, since they don't seem to subject to any observation selection effect; at least from his perspective, his observations were a genuine random sample.

Things get weirder if we consider a variant of the case.

VARIANT: as before, but Charlie has a similar conversation with Bob. Only this time, he tells him he's going to introduce Bob to someone who observed exactly 75 of 100 fish to be big.

New Question: Now what should Bob and Al think?

Here, things get really weird. By the reasoning that led to the Natural Answer above, Al should be nearly certain that ¾ are big and Bob should be nearly certain that ¼ are big. But that can't be right. They would just be agreeing to disagree! (Which would be especially puzzling, since they have ALL the same information.) The idea that they should favor one hypothesis in particular is also disconcerting, given the symmetry of the case. Should they both be 50/50?

Here's where I'd especially appreciate enlightenment: 1.If Bob should defer to Al in the original case, why? Can someone walk me through the calculations that lead to this?

2.If Bob should not defer to Al in the original case, is that because Al should change his mind? If so, what is wrong with the reasoning in the Natural Answer? If not, how can they agree to disagree?

3.If Bob should defer to Al in the original case, why not in the symmetrical variant?

4.What credence should they have in the symmetrical variant?

5.Can anyone refer me to some info on observation selection effects that could be applied here?