Rationality Revisited: The Sleeping Beauty Paradox

Today is Sunday. Sleeping Beauty drinks a powerful sleep potion and falls asleep.

Her attendant tosses a fair coin and records the result.

The coin lands in Heads. Beauty is awakened only on Monday and interviewed. Her memory is erased and she is again put back to sleep.
The coin lands in Tails. Beauty is awakened and interviewed on Monday. Her memory is erased and she's put back to sleep again. On Tuesday, she is once again awaken, interviewed and finally put back to sleep.

In essence, the awakenings on Mondays and Tuesdays are indistinguishable to her.

The most important question she's asked in the interviews is

What is your credence (degree of belief) that the coin landed in heads?

Given that Sleeping Beauty is epistemologically rational and is aware of all the rules of the experiment on Sunday, what should be her answer?

Thirder Position

There are three cases when Beauty is awakened. When it is (Tails and Monday), (Tails and Tuesday) and (Heads and Monday).

But since all of these situations are indistinguishable, she believes that the current situation is (Heads and Monday) with credence $\frac{1}{3}$ , by virtue of The Principle of Indifference. But since that is the only case where we have heads, she believes that the coin landed in heads with one-third credence.

Halfer Position

Since Beauty knows all the rules of the experimental setup, her credence that the coin landed in heads is $\frac{1}{2}$ before she drinks the potion on Sunday.

Since she receives no new information when awaken, she continues to believe that the coin landed in heads with the same credence.

To quote wikipedia,

The Sleeping Beauty puzzle reduces to an easy and uncontroversial probability theory problem as soon as we agree on an objective procedure how to assess whether Beauty's subjective credence is correct.

However, this is not a problem on Probability theory but the concept of subjective probabilities or credences and whether they are well defined concepts.

As a last note, this is a popular problem that is still debated on the internet and among proffessional philosophers.

I am slightly surprised to see that no one has posted this on Brilliant before.

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Comments

This is related to the matter of reincarnation. Let's say that Sleeping Beauty is awakened for the first time in her memory. She is told and understands that the probability that she is a reincarnation from past lives is exactly half---but that she could have had "a million past lives she was awakened into". Using the so-called "Principle of Indifference", that's about a million possible cases, of which being awakened as a reincarnation jumps to about 999,999 out of a million. How did we get from a probability of "exactly half" to about "999,999 in a million" of her being a reincarnation? Either she's a reincarnation or she is not, with the previously declared probability of half, regardless of how many times she's been awakened and interviewed and told otherwise.

Now let's suppose she is NOT told and nor understands anything about reincarnation. She is awakened, and interviewed. No matter what the interviewer asks or tells her (he can lie too), she cannot form an informed estimate of the probability of actually being a reincarnation. Hence, it's useless to assign any probability to it. Her experience in being awakened [multiple times] and interviewed and "told things" adds nothing new to the math.

Michael Mendrin - 5 years, 5 months ago

I do not understand the phrase 'awakened for the first time in her memory'?

Agnishom Chattopadhyay - 5 years, 5 months ago

This hypothetical scenario bypasses the normal birth and growth of a woman---this special Sleeping Beauty only knows that she's been awakened and has no knowledge of her past. This is to highlight the key points of this argument. Either she has gained consciousness for the first time (imagine that she's a robot with AI that's been turned on for the first time) or she's has had awakenings before "in past lives", but has no memory of them. Most reincarnation scenarios imagine that we would not have any recollection of our "past lives".

The crux of the argument is that being told you could be awakened in a succession of times doesn't help you decide whether or not you are having successive reincarnations.

It's quite another argument to make that "when you are awakened, you could be in any of the times you're awakened with equal likelihood." Then, in that case, when you are awakened, you are far more likely to be experiencing a reincarnation. But that depends on "making it a fact" that all those times of awakening occur with "equal likelihood". But no such representation was made to Sleeping Beauty before she was given the sleep potion in the original scenario as posed.

Moral of the Story: Handle the so-called "Principle of Indifference" with great care. Details matter.

What I find interesting about this problem is that it does have an analogy with the question of "phase space" in entropy theory in statistical mechanics--but that is too much to go into in here.

@Michael Mendrin – To explore these issues a bit more, here's a variation of an old con game played with special cards. A con man shows you 2 cards, one with red on both sides, the other with one red and one green side. He illustrates this by slapping both of them on the table, both of them showing red on top. By flipping them over, he proves to you that the other side of one card is red, and the other card green. He throws the two cards into a hat, and pulls one out and slaps it on the table. It shows red again. He says, "Look, you know that the other side is either red or green with equal probability", and so he offers you better than even odds that the other side is green. But the actual probability that the other side is green is $\frac{1}{3}$ , not half. What you don't know is that had he slapped down a card with green showing up, he would have gone into another spiel and find a different way to relieve you of your money.

Try applying the "Principle of Indifference" here.

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$