Variations on the Sleeping Beauty

by casebash 4y10th Jan 20161 min read6 comments

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This post won't directly address the Sleeping Beauty problem so you may want to read the above link to understand what the sleeping beauty problem is first.

Half*-Sleeping Beauty Problem

The asterisk is because it is only very similar to half of the sleeping beauty problem, not exactly half.

A coin is flipped. If it is heads, you are woken up with 50% chance and interrogated about the probability of the coin having come up heads. The other 50% of the time you are killed. If it is tails you are woken up and similarly interrogated. Given that you are being interrogated, what is the probability that the coin came up heads? And have you received any new information?

Double-Half*-Sleeping Beauty problem

A coin is flipped. If it is heads, a coin is flipped again. If this second coin is heads you are woken up and interrogated on Monday, if it is tails you are woken up and interrogated on Tuesday. If it is tails, then you are woken up on Monday and Tuesday and interrogated both days (having no memory of your previous interrogation). If you are being interrogated, what is the chance the coin came up heads? And have you received any new information?

Double-Half*-Sleeping Beauty problem with Known Day Variation

EDIT: This problem should have said: As above, but whenever you are being interrogated you are told the day. You may wish to consider this problem before the above one.

Sleeping Couples Problem

A man and his identical-valued wife have lived together for so many years that they have reached Aumann agreement on all of their beliefs, including core premises, so that they always make the same decision in every situation.

A coin is flipped. If it is heads, one of the couple is randomly woken up and interrogated about the probability of the coin having come up heads. The other is killed. If it is tales, both are woken up separately and similarly interrogated. If you are being interrogated, what is the probability that the coin came up heads? And have you received any new information?

Sleeping Clones Problem

A coin is flipped. If it is heads, you are woken up and interrogated about the probability of the coin having come up heads. If it is tails, then you are cloned and both copies are interrogated separately without knowing whether they are the clone or not. If you are being interrogated, what is the probability that the coin came up heads? And have you received any new information?

My expectation is that the Double-Half Sleeping Beauty and Sleeping Clones will be controversial, but I am optimistic that there will be a consensus on the other three.

Solutions (or at least what I believe to be the solutions) will be forthcoming soon.

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