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Misha comments on Iterated Sleeping Beauty and Copied Minds - Less Wrong
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My instinct is actually that my subjective anticipation for the experiences of every future copy psychologically continuous with myself is 1... for all of them. If I know I am going to be copied 100 times, with one new copy activated every day for the next 100 days how could the 100th copy of me possibly be surprised to learn it was the 100th copy? It has no new information- it knew going into the experiment that there would be some psychologically continuous copy released 100 days from then- that it is that copy isn't new information at all. I have no way to mathematize this intuition but I strongly suspect using a [0,1] density for anticipating multiple copies is just wrong (which doesn't mean it won't work for a large fraction of cases).
If I copied you 100 times, then asked each copy, "Which copy are you?" could you answer? If not, then how can you not gain information when told that you are the 100th copy?
This is still relying on a certain degree of ambiguity. If we are talking about "asking each copy" then to be clear we had best write "how can that copy not gain information when". Each individual copy gains information about identity but there is a 'you' that does not.
That was ambiguously said, yes. How abut this?
The information you-0 start out with that "you will become the 100th copy" is distinct from the information you-100 (or for that matter, you-1 through you-99) gains about identity. It is a lot like the information "someone will win the lottery."
In a sense you-0 should assign probability 1 to being told "You are the 100th copy." In another sense you-0 should assign probability 1/100. This is not a philosophical matter, but a matter of language. We could reproduce the same "paradox" by holding a 10-person lottery between 10 LessWrong users, and asking "What is the probability a LessWrong user wins the lottery?" Here the ambiguity is between "any user" (which happens with probability 1) and "any given particular user" (which happens with probability 1/10).
I think there is room to ask about two probabilities here. If there is something in the future that can only be done by you-42, it will certainly get done, so in this case the probability that you will be the 42nd copy is 1. If I ask each you-1 through you-100 to value a $100 bet that it is the 42nd copy, Dutch Book style, then each should pay $1 for the bet, so in this case we're looking at the 1/100 probability.
You don't have to call these events anything like "You have a 42nd copy" and "You are the 42nd copy". I believe this is a natural description. But in any case, what matters is that there are plainly two distinct probabilities here, and it matters which you use.