*Bounty: 50*

*Bounty: 50*

Consider the following classic problem:

Some researchers would like to put Sleeping Beauty to sleep on Sunday. Depending on the secret toss of a fair coin, they will briefly awaken her either once on Monday (Heads) or twice (first on Monday then again on Tuesday) (Tails). After each waking, they will put her back to sleep with a drug that makes her forget that awakening, and finally she will be awakened on Wednesday without being asked any questions and the experiment will end. When she is awakened (before Wednesday—and she will be told it is before Wednesday, but not whether it is Monday or Tuesday), to what degree should Sleeping Beauty believe that the outcome of the coin toss was Heads?

In a previous thread (where I borrowed and slightly modified the quotation), whuber convincingly argues that the problem as stated above is ambiguous and gives interpretations under which the answer is either $frac{1}{3}$ or $frac{1}{2}$, with $frac{1}{3}$ being the more interesting answer. I recommend reading whuber’s response before attempting to respond to this post.

Now consider the following modification, borrowed from a 2015 blog post:

Before going to sleep on Sunday, Sleeping Beauty makes a bet at odds of 3:2 that the coin will come down heads. (This is favourable for her when the probability of heads is 1/2, and unfavourable when the probability of heads is 1/3). She is told that whenever she is woken up, she will be offered the opportunity to cancel any outstanding bets. Later she finds herself woken up, and asked whether she wants to cancel any outstanding bets. Should she say yes or no? (Let’s say she doesn’t have access to any external randomness to help her choose). Is her best answer compatible with a “belief of 1/3 that the coin is showing heads”?

The issue in the modified version is that because the coin is fair, the expected value of the bet should be $3 cdot frac{1}{2} – 2 cdot frac{1}{2} > 0$. But when Sleeping Beauty is awoken, by whbuer’s reasoning, she assigns a probability of $frac{1}{3}$ to the coin coming coming up Heads given that she is the one awakened. In this case, the expected value is $3 cdot frac{1}{3} – 2 cdot frac{2}{3} < 0$, so she should cancel the bet. Yet nothing about the bet seems intuitively to have changed.

In the blog post linked above, Sleeping Beauty reasons that while the probability she assigns to the coin coming up Heads based on being awakened before Wednesday is $frac{1}{3}$, on Wednesday she will experience the event of waking up on a Wednesday, at which point the probability of Heads given waking up on Wednesday will be $frac{1}{2}$, so she defers the bet to later.

However, since Sleeping Beauty already knows earlier that she will eventually wake up on Wednesday, doesn’t that argument mean that the answer to the original Sleeping Beauty paradox should “morally” be $frac{1}{2}$ rather than $frac{1}{3}$? How do you resolve the clash between the intuitive feeling that Sleeping Beauty should not cancel the bet and whuber’s reasoning for the probability of Heads given that she was the one awoken being $frac{1}{3}$? Should Sleeping Beauty cancel her bet?