Actually, you're right about the infinite series of bets. Let N be the number of times Sleeping Beauty is to be woken up. Suppose (edit: on each day she wakes up) Sleeping Beauty is offered the following bets:
In each individual bet, the second option has an infinite expectation, while the first has a finite expectation. However, if Sleeping Beauty accepts all the first options, she gets $10 every day she wakes up, for a total of $10N; if Sleeping Beauty accepts all the second options, she gets less than $2 every day she wakes up, for a total of $2N. Even though both options yield infinite expected money, this is still clearly inferior.
I suspect though, that this is a problem with the infinite nature of the experiment, not with Sleeping Beauty's betting preferences.
That's not what I meant. I meant... ugh I'm really tired right now and can't think straight.
maybe:
Pot starts at 1$, each iteration she bets the pot against adding one dollar to it if N is greater than the number of iterations so far, with if needed the extra rule that if she gets woken up an infinite number of time she really gets infinite $.
To sleep deprived to check if the math actually works out like I think it does.
It's well known that the Self-Indication Assumption (SIA) has problems with infinite populations (one of the reasons I strongly recommend not using the probability as the fundamental object of interest, but instead the decision, as in anthropic decision theory).
SIA also has problems with arbitrarily large finite populations, at least in some cases. What cases are these? Imagine that we had these (non-anthropic) probabilities for various populations:
p0, p1, p2, p3, p4...
Now let us apply the anthropic correction from SIA; before renormalising, we have these weights for different population levels:
0, p1, 2p2, 3p3, 4p4...
To renormalise, we need to divide by the sum 0 + p1 + 2p2 + 3p3 + 4p4... This is actually the expected population! (note: we are using the population as a proxy for the size of the reference class of agents who are subjectively indistinguishable from us; see this post for more details)
So using SIA is possible if and only if the (non-anthropic) expected population is finite (and non-zero).
Note that it is possible for the anthropic expected population to be infinite! For instance if pj is C/j3, for some constant C, then the non-anthropic expected population is finite (being the infinite sum of C/j2). However once we have done the SIA correction, we can see that the SIA-corrected expected population is infinite (being the infinite sum of some constant times 1/j).