I agree with the author of this article. After having done a lot of research on the Sleeping Beauty Problem as it was the topic of my bachelor’s thesis (philosophy), I came to the conclusion that anthropic reasoning is wrong in the Sleeping Beauty Problem. I will explain my argument (shortly) below:
The principle that Elga uses in his first paper to validate his argument for 1⁄3 is an anthropic principle he calls the Principle of Indifference:
“Equal probabilities should be assigned to any collection of indistinguishable, mutually exclusive and exhaustive events.”
The Principle of Indifference is in fact a more restricted version of the Self-Indication Assumption:
“All other things equal, an observer should reason as if they are randomly selected from the set of all possible observers.”
Both principles are to be accepted a priori as they can not be attributed to empirical considerations. They are therefore vulnerable to counterarguments...
The counterargument:
Suppose that the original experiment is modified a little:
If the outcome of the coin flip is Heads, they wake Beauty up at exactly 8:00. If the outcome of the first coin flip is Tails, the reasearchers flip another coin. If it lands Heads they wake Beauty at 7:00, if Tails at 9:00. That means that when Beauty wakes up she can be in one of 5 situations:
Heads and Monday 8:00
Tails and Monday 7:00
Tails and Monday 9:00
Tails and Tuesday 7:00
Tails and Tuesday 9:00
Again, these situations are mutually exclusive, indistinguishable and exhaustive. Hence thirders are forced to conclude that P(Heads) = 1⁄5.
Thirders might object that Beauty’s total credence in the Tails-world would still have to equal 2⁄3, as Beauty is awakened twice as many times in the Tails-world as in the Heads-world. They are then forced to explain why temporal uncertainty regarding an awakening (Monday or Tuesday) is different from temporal uncertainty regarding the time (7:00 or 9:00 o’clock). Both classify as temporal uncertainties within the same possible world, what could possibly set them apart?
An explanation could be that Beauty is only is asked for her credence in Heads during an awakening event, regardless of the time, and that such an event occurs twice in the Tails-world. That is, out of the 4 possible observer-moments in the Tails-world there are only two in which she is interviewed. That means that simply the fact that she is asked the same question twice is reason enough for thirders to distribute their credence, and it is no longer about the number of observer moments. So if she would be asked the same question a million times then her credence in Heads would drop to 1/1000001!
We can magnify the absurdity of this reasoning by imagining a modified version of the Sleeping Beauty Problem in which a coin is tossed that always lands on Tails. Again, she is awakened one million times and given an amnesia-inducing potion after each awakening. Thirder logic would lead to Beauty’s credence in Tails being 1/1000000, as there are one million observer-moments where she is asked for her credence within the only possible world; the Tails-world. To recapitulate: Beauty is certain that she lives in a world where a coin lands Tails, but due to the fact that she knows that she will answer the same question a million times her answer is 1/1000000. This would be tantamount to saying that Mt. Everest is only 1m high when knowing it will be asked 8848 times! It is very hard to see how amnesia could have such an effect on rationality.
Conclusion:
The thirder argument is false. The fact that there are multiple possible observer-moments within a possible world does not justify dividing your credences equally among these observer-moments, as this leads to absurd consequences. The anthropic reasoning exhibited by the Principle of Indifference and the Self-Indication Assumption cannot be applied to the Sleeping Beauty Problem and I seriously doubt if it can be applied to other cases...
I don’t think you fully understand my argument. It is not about being offered a wager or not, because that certainly would alter the experiment and make it very easy to decide whether halfer or thirder reasoning is the way to go.
Instead, it is about the fundamental principle the thirder’s argument is based on; the anthropic principle Elga calls his Principle of Indifference. It is the key element used to justify Beauty’s credence drop from 1⁄2 to 1⁄3 on waking up. This credence drop is in serious need of justification because Beauty learns nothing new when she wakes. She only learns ‘Today is Monday or Tuesday’ which she knew she would learn beforehand. That is, she receives no knew information on which she can conditionalise. Therefore thirders resort to anthropic reasoning, which goes like this: “I am in one of three awakenings now, which all look the same to me. Therefore I should didvide my credence equally over them.”
My counterargument tries to show the fallacy of this reasoning by creating two other possible awakenings within the Tails-world. Hence there are then 4 possible awakenings within the Tails-world and thirders adhering to the Principle of Indifference should divide there credence equally over them. If they don’t, then that means that it is not about Beauty’s number of observer-moments within a possible world, but about the number of times Beauty is asked the same question.
Like you pointed out, Beauty is still awakened twice if Tails and once if Heads. Therefore she is indeed vulnerable to being Dutch Booked. The problem with the wager you proposed is that it is repeated twice if Tails and once if Heads, which makes it unfair. Suppose someone offered you a bet that paid 10$ if a coin comes up Heads and cost you 1$ if the coin comes up Tails. The catch is; if the coin comes up Tails the bet is repeated 100x times. Clearly you do not accept this bet, as the real bet is one where you stand to lose 100$ instead of 1$. However, this changes nothing about your belief that the objective chance of a coin to land Heads is 1⁄2. Beauty will not accept any bets that are repeated if lost. Dutch Book arguments in the Sleeping Beauty Problem are inconclusive since they are imaginable for both thirders and halfers. Hence they do not provide any deeper insights into the halfer and thrider arguments.
PS I’m sorry if I came on too strong; it was my first post here at LessWrong and I’m still reading my way through all the articles.