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.

]]>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.

]]>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).

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