Sleeping Beauty Paradox

[bigmonkey]

Hi all. I've been hanging around this forum for a while and am surprised at the intelligence and ranges of knowledge on display. So I thought I'd pose the Sleeping Beauty Paradox for people to have some fun with. I've read nearly every published paper on this and I think I've cracked it, but I did go through a period where I thought I'd cracked it several times only to be reduced to quivering confusion, so I can't be entirely confident in my own answer. Anyway here is the paradox. I thought about posting a poll but reasoning is more interesting than intuition.

Sleeping Beauty is a paragon of rationality, a perfect Bayesian, entirely rational etc...etc...It is Sunday evening now. We are going to put Beauty to sleep and then flip a fair coin. If the coin lands heads then we will wake Beauty up on Monday evening, then she will go back to sleep until next week, when the experiment will be over. If the coin lands tails we will wake her up on Monday evening, put her back to sleep, and wake her up on Tuesday evening, then she will sleep again until after the experiment. Furthermore after she wakes up on Monday (she will wake up on Monday), we will administer her with an amnesiac which will make her forget that awakening. Whenever she wakes up she won't know what day it is. Suppose also that before the experiment she knows exactly what the experience of waking up ignorant of the date will be like. She won't learn anything upon wakening. Beauty knows all of the above.

On Sunday evening we ask Beauty what she thinks the probability is of the coin landing heads. Unsurprisingly she says 1/2. Whenever she wakes up she will say 1/3. Furthermore, before the experiment starts she knows that upon waking up she will say that the answer is 1/3. Before the experiment starts then she is in a position such that she believes the answer is 1/2, but at a later point without learning any new information and without becoming irrational at any point, she will believe the answer is 1/3. She is said to break Bas van Fraassen's principle of reflection which is intuitively very appealing as a rationality constraint.

A brief argument for 1/3: she knows that the chance of heads is 1/2, and heads results in one awakening. The chance of tails is 1/2 and tails results in two awakenings. Therefore over the time period of the experiment she expects to wake up 1.5 times, once a tails-awakening and half a time a heads-awakening, on average. Therefore when asked how likely was it that the coin landed heads she will answer 1/3, and tails 2/3, conditionalizing on the evidence that she is awake, (although she always knew she'd wake up and exactly how that would feel) She knows that a heads-awakening is half as likely to her happen than a tails-awakening.

So what's the solution? Do we throw out Reflection? Does she believe 1/2 when she wakes up too?

[jogger08152]

She will wake up all Mondays regardless of the flip, and half of Tuesdays (when the result is tails).

Since...

A) upon wakening, she has no idea what day it is and
B) she does not recall whether she has been awake the previous night,

...the wakening itself offers her no additional insight into the results of the flip, and she therefore has no reason to change her assessment of the probability of either result from 1/2.

[bigmonkey]

OK I should've phrased the question a bit better. Before she sleeps it's obvious that the rational answer is 1/2, no argument there. But after she wakes up it seems there are two equally valid arguments, one saying 1/2, one saying 1/3. You've made the valid argument that probabilities don't change when no new evidence has been received, but the most disconcerting part of the paradox is that the answer of 1/3 also seems correct. It looks like it should be a different way of working out the same answer but it gives us a different answer instead, leading to paradox.

Is your position that, the answer is definitely 1/2, therefore it cannot be 1/3, therefore the reasoning for 1/3 is somehow wrong. If that's your position then how is the reasoning for 1/3 invalid?

[KipBond]

[ QUOTE ]
Sleeping Beauty is a paragon of rationality, a perfect Bayesian, entirely rational etc...etc...
...
On Sunday evening we ask Beauty what she thinks the probability is of the coin landing heads. Unsurprisingly she says 1/2. Whenever she wakes up she will say 1/3.

[/ QUOTE ]

The statement that she wakes up & says "1/3" is not part of the problem, right? Because if so, there's a contradiction in the problem. When she is woken up, and asked what the probability is that the coin was heads, the rational response is "1/2" not "1/3".

The fact that you get her to answer twice if it's tails, and answer only once if it's heads doesn't change the probability that the coin was heads. If she has to pick an outcome, though, she will be right 2/3 of the time she is asked by answering "tails".

[bigmonkey]

[ QUOTE ]
[ QUOTE ]
Sleeping Beauty is a paragon of rationality, a perfect Bayesian, entirely rational etc...etc...
...
On Sunday evening we ask Beauty what she thinks the probability is of the coin landing heads. Unsurprisingly she says 1/2. Whenever she wakes up she will say 1/3.

[/ QUOTE ]

The statement that she wakes up & says "1/3" is not part of the problem, right? Because if so, there's a contradiction in the problem. When she is woken up, and asked what the probability is that the coin was heads, the rational response is "1/2" not "1/3".

The fact that you get her to answer twice if it's tails, and answer only once if it's heads doesn't change the probability that the coin was heads. If she has to pick an outcome, though, she will be right 2/3 of the time she is asked by answering "tails".

[/ QUOTE ]

No her saying it is 1/3 is not part of the problem. The fact that there is a rational argument for 1/3 is the problem, and Beauty being rational we would expect her to get it.

I definitely agree with you about the being right twice idea. Well she is never right because she says either 1/3 or 1/2, and the answer is always either 0 or 1. But if she is making bets on the outcome of the coin, she has to believe 1/2 before she sleeps and 1/3 when she wakes up otherwise she will be Dutch-booked for sure.

Suppose she believes 1/2 the whole way. Pre-sleep she bets $15 on tails to win $30, and post-sleep she bets $10 on heads to win $20. If heads she loses her first bet, wins her second bet and is down $5. If tails she wins her first bet then loses her second bet twice, losing $5 no matter what happens.

If she believes 1/2 pre-sleep and 1/3 post-sleep she bets $15 on tails pre-sleep to win $30 and $10 on heads post-sleep to win $30. If heads she loses the first bet and wins the second winning her $5. If tails she wins the first bet and loses the second twice losing her $5 and making the bets technically fair.

So it would seem that if Beauty is a rational Bayesian she must believe 1/2 pre-sleep and 1/3 post-sleep. It is also said that Bayesians argue that a rational agent's credence in a proposition is equivalent to the odds they are willing to accept bets on it, suggesting that she really does believe 1/2 pre-sleep and 1/3 post-sleep, and that she knows all this in advance.

[KipBond]

This is really no different than doing 1 coin flip, if you say "heads" you get 1:1 payout, if you say "tails" you get 2:1 payout. Of course choose "tails", but the odds that you will win is still 1/2 -- you just win twice as much if you choose "tails" instead of "heads".

[bigmonkey]

[ QUOTE ]
This is really no different than doing 1 coin flip, if you say "heads" you get 1:1 payout, if you say "tails" you get 2:1 payout. Of course choose "tails", but the odds that you will win is still 1/2 -- you just win twice as much if you choose "tails" instead of "heads".

[/ QUOTE ]

It is very much like this. It's more like a case where you make a standard and fair bet on heads, on the condition that if you win you must make two bets that are guaranteed to lose worth $x each, and if you lose the first heads bet you are given $x back. The original "game" is only worth playing if you demand that if you lose the first bet on heads you get given $1.5x back.

The conclusion I've come to is that the Bayesian theory that degrees of belief in propositions are equivalent to the worst betting odds one is disposed to accept on the proposition being true, is simply false.

I'm a little surprised we got to the crux of the paradox so quickly. I think it may be because I left out a really good argument for 1/3 in the first place offered by Adam Elga. Also I didn't think anyone would be willing to give up that Bayesian theory as it's pretty firmly established although quite clearly false as this example shows.

[Siegmund]

I'll be interested in seeing what good argument, if any, there is for 1/3... sure haven't heard one yet and have trouble imagining one yet.

Cliff notes: she knows she goes to sleep with 1/2 chance of this happening, and she knows that when the experiment is over, she's going to only remember waking up once, regardless of whether she really woke once or twice. No information, no change to the prior, wtp?

[AWoodside]

P(H) = 1/2
P(Being awake) = 1
(Being awake | H) = 1
P(H | Being awake) = P(Being Awake | H)*(H) / (Being Awake) = 1 * (1/2) / 1 = 1/2

Is this wrong? Isn't this the standard way to do Bayesian probability? I don't understand how she gets 1/3.

[f97tosc]

I agree that the rational answer is 1/2 before, 1/3 during, and 1/2 after the experiment. Certainly if she was offered wagers every time she woke up (and we repeated the setup many times), she would lose money if she started betting on any other odds.

It may be tricky to pinpoint exactly what she has learned when she is awakened, but she should be aware that the test protocol has introduced a bias in favor for heads, and that, under different experimental outcomes, she might have been sleeping and not in a position to state her belief.


I can't say I am familiar with the reflection principle, but I suspect that the "resolution" will have to do with the fact during certain experimental outcomes, we supress Sleeping Beauty's ability to state (or even form) her belief.