Sleeping Beauty Meets Monty Hall
Sleeping Beauty participates in the following experiment. On Sunday she is put to sleep, and a fair coin is flipped. Regardless of the result of the coin flip, she is awakened on Monday and asked whether she thinks the coin was heads or not. If the coin was tails, however, then she is put back to sleep with her memory erased, and awakened on Tuesday and asked the same question again. She knows the protocol. She is awakened one morning and instead of the expected questions she is offered a bet. She may pay $600 in which case she will get $1000 if the coin was tails. Should she take the bet?
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Joseph:
Given that the people running the experiment were willing to offer this bet, I’d say there’s a 100% chance the coin was heads.
27 August 2011, 8:57 amzplesivcak:
What use is to win a bet if you don’t remember that you won it. 😉
27 August 2011, 10:11 amtr1n:
Nice use of phrasing. I’d have to say no, only because if it is tails she will be put back to sleep with her memory erased, thus losing the $1000. If it is heads, well, she is out of $600. Lose lose situation.
27 August 2011, 10:47 amBill:
She should not take the bet.
She knows the protocol, but it is changed on her after she wakes up. She has no way of knowing whether the bet would have been offered to her regardless of the outcome of the coin flip. And, as Joseph points out, it’s a little suspicious of them to offer her the bet after the fact. Also, even if she knew the bet would be offered either way, she’d also have to know whether or not they were going to offer her the bet twice if the result were tails. But she does not have this information.
She should not take the bet even if asked to risk $100 against $1000. She has no reasonable way to compute her odds.
In the Monty Hall problem, the contestant knows the protocol ahead of time and it is not changed. This is a crucial detail in determining the solution (and leaving this detail out is a common but decisive mistake in presenting the problem).
27 August 2011, 12:01 pmPeter Gerdes:
Like the Monty Hall problem without a precisce specification of the problem there is no correct answer.
For instance if Monty Hall only chooses to open another door if he knows you’ve picked the door with the prize you obviously shouldn’t switch.
Similarly in this case we need to specify whether or not Sleeping Beauty is offered this bet every time she is woken up. I take it from your description this is what you intend. Once this is specified the answer is trivially calculated.
Sleeping Beauty’s expected return from following the strategy of taking the bet whenever woken is as follows
P(Heads)*-600 + P(Tails)*(400 + 400) = 100
Obviously her expectation if she follows the strategy of not betting is 0 so yes (absent any risk aversion etc..) she should take the bet.
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Indeed, phrasing the problem this way exposes the fact that ‘paradox’ in the sleeping beauty problem is merely a result of asking a vague, ill-defined question. The problem is taking probability to be some kind of objective thing out there in the world but probability isn’t like electric charge or some other physical quantity out in the world. Probability is nothing more than a fancy name for counting (or doing measure theory). You can’t have a paradox about what the probability of some event because to even talk about probability you have to have already reduced the problem to a straightforward mathematical specification.
Sleeping Beauty goes wrong because what is really being asked is something like ‘what should Sleeping Beauty believe about the coin flip’ and there is no actual set of events over which a probability measure is defined. It’s only a paradox because we confuse probability and some vague and incoherent notion of rational belief. Your example demonstrates this by showing that as soon as we look at well defined properties of the event space the answer becomes clear.
29 August 2011, 9:38 amPeter Gerdes:
Just to emphasize I was answering the question of which strategy would be most beneficial for Sleeping Beauty to follow.
There is a different question of what strategy is it rational for sleeping beauty to adopt but to even get started answering this question one would need to make clear what she knows before going into the experiment.
29 August 2011, 9:40 amTanya Khovanova:
I hoped that my readers notice that the problem is ambiguous. Like in Monty Hall the answer depends on the strategy of the host. Is she offered the bet once per awakening, or once per flip or only if the coin is heads?
29 August 2011, 10:24 am