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.
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.