Games - Sleeping beauty problem

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	Games - Sleeping beauty problem

The sleeping beauty problem is a puzzle in probability theory: a sleeper is to be woken once or twice according to the toss of a coin, and asked her credence for the coin having come up heads.

Statement of the problem

The problem is this:

Beauty is to be put to sleep and we secretly toss a coin.
We subseqently wake her up on Monday.
We wake her again on Tuesday if the coin came up tails.
Each time after waking her we ask: What is your credence now for the proposition that the coin came up heads?
She will be unable to distinguish the awakenings, in particular forgetting the first if awakened a second time.
She knows these details; how should she answer?

The days of the week may seem irrelevant, but are included because they are used in some expositions.

Proposed solutions

There are two schools of thought, the "thirders" and "halfers", named after the answer they suggest she should give. Yet others say that there are reasonable definitions of credence making either answer correct.

It is in any case clear that if required to bet she would need to assign heads a probability of 1/3 in order to break even.

Thirder arguments

Thirders may for example argue that from her point of view all three awakenings are equally likely, and that only one of them means the coin came up heads. They may also argue that if she were not to be woken at all on heads, she would give it a credence of 0 when woken, invalidating any halfer argument.

Halfer arguments

A typical halfer argument is that the probability that her awakening provides no new information, and so her credence must remain 1/2.

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