I'm not sure about the logic in this paragraph. A "trial" as you are stating it from the point of view of the experiment-runner, not Sleeping Beauty. But the question is asked from her perspective.

When the coin goes heads, she is only woken once and the number of experimenter-trials and SB-trials are in agreement. When it goes tails, she is woken twice and asked the question twice. From her perspective, 2 trials have happened. They aren't independent but she doesn't know that.

Anyway my brain is starting to hurt so I have to stop thinking about this for a while.

What prevents the sleeping beauty once she knows the entire story to actually step out of herself so to speak and become an external observer and indeed realize that she is awaken 2/3 of the total times on Mondays and 1/3 on Tuesdays resulting in the 2/3 chance for tails. I mean if she bets for tails 1-1 every time she is asked the question and we bet against it we will lose . Can one argue against this? She will be asked 3/2N times for N trials and N/2 of them it will be heads and N=N/2+N/2 tails. So she will come ahead by expecting tails to have happened more often. Can one seriously debate that this is not instantly an incentive for her to believe that its tails more often when asked the question?

The secret here is that she has no memory, but she knows she has no memory and that she is exploited multiple times on tails meaning more often than not it will be tails that caused her waking up either the first or the second time ! 1 tail causes 2 events of waking up. 1 head causes only 1.

You do not disagree with this so why dont you agree it has to be 2/3-1/3 and not 50-50 for her way of seeing things.

I seriously fail to see why this has attracted so much attention. Exactly what is the 50% camp claiming, that she should expect when asked the question that tails caused it 50% of the time? WHY? Each tail causes multiple questions.

She has no problem of course agreeing that the coin is a 50-50 coin. But it is the fact she faces a waking up that is the issue. A random wake up will tend to be 2/3 of the time the result of a tail flipped. Her problem, is that she doesnt know what day it is. Its very easy to visualize this by taking the large number of pills limit. Imagine her spending her entire life waking up due to a tail day after day. How would she dare bet for heads as 50-50?
In the end her aging would be telling the story of course!

The bolded is the crux of the argument by 1/3'ers. You are appealing to the indifference principle - which we know can be problematic. Using Bostrom's notation H1,T1,T2 where H1 is heads and awakened Monday, T1 is tails and awakened Monday, T2 is tails and awakened Tuesday. You are saying that when Beauty is awakened she should be indifferent to the 3 possibilities H1,T1, and T2 and thus give equal credence to each. But you have no proof for this and as we well know, just because there are 3 possibilities doesn't mean they are equally likely.

To my mind it is more reasonable for Beauty to decide she is in one of two equally likely possible worlds, either the world where heads was flipped or the world where it was tails. Furthermore, to avoid contradictions, I think I'm agreeing with jason1990's innovation by claiming she should believe it is either heads and Monday or tails and she can form no credence for it being either Monday or Tuesday. This last "noncredence" is because any credence she might form leads to contradiction.

If on being informed it's Tails Beauty were to appeal to the indifference principle and form a 50-50 credence for Monday-Tuesday and if her credences are to act like probabilities, then if she is awakened and informed it is Monday she should form a credence of 2/3 for Heads. But this is problematic because the Coin might just as well be flipped after she is put back to sleep on Monday as on Sunday night. After all, regardless of the coin flip she is going to be awakened on Monday. But it certainly makes no sense for her to predict a coin flip yet to take place is going to be heads with 2/3 credence when she is told it is Monday.

I don't think forcing Beauty to bet on heads proves anything. She knows that she will be forced to bet twice as many times when she's wrong so naturally she will want odds. If I'm going to bet on a black card in a random fair deck I'm going to demand odds if I know I'll be forced to bet twice as much when the card is red.

I suggest you read at least the last 100 posts or so in the thread I linked - as usual pay special attention to jason1990's posts. As you can see, we pretty much beat this subject into exhastion in that thread. Also, I think Bostrom's paper that I linked is worth a read. He presents a hybrid model which attempts to somewhat satisfy both the 1/2 and 1/3 point of view. btw, I think he's one of the guys claiming we probably live in a simulation - or something like that.


PairTheBoard

First of all i appreciate you taking the time to respond and produce the links and also for your efforts back then that looks extensive.

However i personally see no reason to go over details of arguments that have no way in hell chance to convince me i am wrong.

In order for my thinking to change i need someone to show me why it fails. I need a crystal clear example of failure.

All i said is that she will always bet on tails and there is nothing we can do about it if we keep score or an independent bank does because she loses memory, she will be informed in the end she is ahead for doing that. Betting for tails and thinking it tends to be tails that caused her waking up are the same thing. Someone has to protect her from the memory loss and she has to become that external observer herself. If she doesnt do that and doesnt know the pill story then she will always think erroneously that its always Monday and conclude 50-50. So what more evidence do i need than the fact that the only way she can arrive at 50-50 is by a lie! Or that the only way to validate that prediction of 50-50 is for us not to give here the pill and therefore behave identically with a 50% coin.


If she chooses tails every time we wake her up and we keep an honest record of all this she will be ahead eventually when a pill is used with a fair coin. She will be as ahead eventually as if there was never a pill involved and we had used an unfair coin that flips tails 2/3 of the time and she knew that. She would render by betting for tails the exact same EV with the 50-50 plus pill coin. That right there settles it in my opinion.


Show me an example where the calculation i originally did using the separation into 2 days and the fact established clearly to an external observer that say videotapes all this and then randomizing the clips picks up one to view without looking at its production date, that is only revealed at the end of the questioning. Isnt he going to always find out that P(Monday)=2/3 and P(Tuesday)=1/3 when viewing these clips? Isnt the questioning of the sleeping beauty an identical matching with one of those clips every day (there is a 1-1 correspondence, but of course only an external observer can tell which is which) . You think i am implicitly using here a principle that comes under question. It is never under question for the only observer that knows the whole truth and this is the external observer. I have to use the viewpoint of the external observer if in the shoes of sleeping beauty because i have no memory of my own.

Any 50-50 camp would basically argue the following absurd thing;

If i have 2 sleeping beauties that in one of them i give only 1 pill and the other i give 1000 pills and i wake them up and always ask them what caused this waking up they should both choose 50-50 and claim its equally likely to be a heads or tails. Obviously the 2 cases N=1 and N=1000 are radically different . How can it be true that they both respond the same way now? This should never happen. Only if they dont know the pill story. Whatever the right answer is , it cant be one that makes the 2 cases identical because they arent.


Of course she splits in 2 equally probable branches of the world on Monday. But in one of them she will be asked multiple times. She knows that and its no different than trusting the picture of the external observer. She can only say 50-50 if the pill is never mentioned.


Seriously show me an argument that forces me into a mistake my using the angle of the external observer in order to judge the probability of what day it is when she wakes up. You call this the indifference principle. I call it the principle of counting the number of wake ups properly. That counting can never be questioned. All i have to ask her when she wakes up for example is what day is it? What is your answer for that? Shouldnt she bet always on Monday?

I dont know what is basically telling you to doubt the analysis that derives the probability for being Monday or Tuesday. I honestly want to see an example of failure in my logic. Otherwise reading all this material is probably going to be a waste of time. I will however gladly do so if offered an example where my thinking leads to an error. Can you honestly not agree with the elementary fact that if she always bets on tails 50-50 she will be ahead in the end? Isnt that equivalent to thinking P(tails|waking up)>P(heads|waking up)? What is the 50% camp saying about the fact in this bet she will be ahead in exactly the same way she would be with a 2/3 unfair coin.


PS: i have gone over the links yesterday of course not all 100 posts but the last few and the linked papers (but didnt real the full length of these papers) and it seems there was no verdict so to speak and lack of confidence in ideas that make very safe sense to me without any convincing argument for this reason of lacking confidence. Hence my hesitation to go over 17 pages of otherwise bright people that i know will not be seeing the world as i do and are not doing something to alter that judging from the way the thread ends...Also the day i will leave a math or physics problem to philosophers is the day i will die...Plus i much rather converse with your or Jason's 2011 version (lol) than 2007. I also prefer the way i think in general today about things than the way my self thought in 2007 (well in some things at least).

It's a 17 page thread (25 posts per page). I asked you to read the last 4 pages. Considering the length of your posts which you expect us to read I don't think that's asking too much. You might consider the fact that this problem has been around for years and opinions among smart people who have studied it for considerable time remains fairly evenly divided. That should tip you off that your attitude that the solution is simple and should be clear to any competent reader of your explanation - arrived at after relatively mininal thought or consideration of other views - might be a bit naive. Reading Bostrom might give you a better idea of some of the professional thinking this problem has provoked.

How can you possibly know this if you haven't studied the arguments? If that's your attitude I don't see much point in repeating them to you here.




The problem with creating a perfect analogy is that crystal clear cases where we are confindent in our bearings do not involve amnesia. That's the problem here. How should we handle her agency under amnesia. I gave the best analogies I could in the other thread.


As I said before, If she is forced to bet on heads she will naturally demand odds because she knows she will be forced to bet twice when she is wrong.

jason1990 emphasizes this point as crucial and I think he is right. The coin is flipped before she goes to sleep Sunday night but she's not told the outcome. When she goes to sleep Sunday night she has the information that the coin has 50% chance of being heads. ie. She knows she nows lives either in a world where the coin landed heads or tails. Her credence for each is 50%. She also knows she will be awakened not knowing which world she is in, a heads world or a tails world.

Here's the crucial point. When she is awakened she has not recieved any additional information. Therefore her credence for which world she is in should remain 50-50. Arguments regarding this point are probably what you should study in the 2007 thread - last 4 pages or so. Having received no additional information from her awakening, when she is awakened she retains her prior credence for 50% that's she's in the heads world. But if asked to bet on heads, knowing she will be asked twice when wrong, she will demand odds.

The closest analogy I could come up with is two bags. One with 1 black ball. One with 100 white balls. A coin will be flipped and Alice is told if it's heads she will find herself awakened as a ball in the 1st bag. If tails she will find herself awakened as a ball in the 2nd bag. The coin is flipped and Alice awakes. What is her credence that she is a black ball? What is her credence she is a white ball?

Suppose she is told she will have no short term memory and no sense of time while being a ball. And that she will spend 1 week if in the 1st bag but only 1 hour if in the 2nd bag. Should her credence now be different when she finds herself being a ball in a bag?

Suppose instead she spends a week in each bag but if she's in the 2nd bag she will randomly switch from being one white ball to another white ball every 5 seconds - but she will have no awareness of the switch. Being one white ball is the same as being another for her. Finding herself a ball in a bag should she now have yet another credence for whether she is a white or black ball? I don't think so.

In all cases, finding herself being a ball in a bag has provided her with no new information. In each case her credence should be 50% that she's a black ball based on what she knows before becoming a ball. Of course if she is required to bet (no short term memory) on it every second, or every 5 seconds, or every day, or every time she unknowingly becomes a different white ball, she will demand odds according to the situation. But that doesn't change her credence for the color ball she finds herself being.


PairTheBoard

PairTheBoard,

2+2=4 lol

Only making sure here we answer the same thing;

Basically this thread for me is about this question;

She is awaken one day without knowing which after an event that took place on Monday regarding a coin tossing that had 50% chance to be heads/tails (fair coin). Given the story with the pill when its tails and the repeated waking up with memory loss on Tuesday for the case of N=1 pill, this thread now asks her this upon waking up;

What do you think is the chance that the coin was tails when it was flipped on Monday?

My answer to this question is 2/3 .

Real simple, is your answer 50%? Is your answer 2/3? Is it true you have no answer?

She should answer, "50% but if I'm to bet on heads I need odds because I'm being settup to bet twice when I'm wrong."


I don't think I'll respond further unlesss and until you have addressed the points I've made in this thread and shown some knowledge of the final arguments in the 2007 thread. I don't see the point in your continuing to repeat the same argument over and over. Counterpoints are available here and in the 2007 thread if you have any interest in them. If you don't I see no reason to respond further since the only response you seem to care about is one that simply agrees with you.


PairTheBoard

I think this is great read.
Very interesting problem btw, I didn't hear about it before. My gut reaction was "50%, wtf, no new information from awakening" and it took me a while to see the point of 1/3 camp.

PairTheBoard real simple;

SB is woken up N>>1 times in her life after participating for years in similar tests and instead of saying anything she asks them with confident attitude; "was it tails ?". They respond to her how many times with a yes? Call it M. Since she has no memory i will keep the record for her because i am her lover. I will remain faithful to her and to record keeping until the end of time.

Isnt after many years N->inf the probability of tails the limit ratio M/N? Can this ratio be far from 2/3? (1/3 for heads)

Every time she is awaken and interviewed is identical to all prior ones. She has no memory and each interview is a new experience. Are you suggesting the 2 tail interviews should count as 1? For her they remain 2 distinct interviews although of course they are correlated. If you find yourself awake one day with them in front of you you better guess it was tails!

Enough said.

Its not 50% but i need odds, it has another name , its called 1/3 (or 2/3 in my way of dealing with tails rather than heads).

I think I have a nonzero credence that I've figured it out. Jason's ppqqq space is annoying, in that it is consistent by definition, but in addition to allowing 1/2 -> 1/3, it appears you can formulate spaces that allow 1/2 -> 4/5 or any other manner of 0<silliness<1. I think I have found a clear reason to reject all but one family of spaces as relevant to the problem.

There are a couple of observations that I think everybody agrees with. If you run this experiment a large number of times, and then "draw" AN AWAKENING at random, it will be the result of tails 2/3 of the time. Also, if you draw AN ENTIRE EXPERIMENT (the full sunday-wednesday) at random, then look inside at any awakening to see which kind it is (HM/TM/TT), it will be a tails-experiment only 1/2 the time. So it seems like a question of how to sample.

So let's use jason's space, modified slightly to be ppxyz (x=HM, y=TM, z=TT). First of all, what does this space represent? If an observer selects an experiment, then peers inside to reveal one of these five parts, this is the probability that he will observe a given part. We have the 2 ps just to make a pre-sleep credence 1/2 as it should be. So when we draw from this space, conditional on not getting a p, what happens? The 1/3 crowd says (3 choices, indifference principle, x=y=z=(1-2p)/3, heads 1/3, gg).

This makes no sense to me. Since one observation determines the experiment to be a H/T experiment (you can't get a TM in a heads experiment obviously), if our observer from before uses the x=y=z probability space, then he believes he will only observe a heads-experiment 1/3 of the time. But we already agreed that he will actually observe a heads-experiment half the time. So the x=y=z space should be rejected. To not be rejected for this reason, x=.5(1-2p) is the only option. And then y=z=.25(1-2p) is the natural followup.

Is there a contradiction with a single-experiment probability space of .5,.25,.25 awakenings and a long-term average of 1/3,2/3 awakenings? No. Because every time you sample TM or TT, you know the other one occurred in the experiment. So if you sum awakenings over experiments, you get the appropriate 1:2 ratio.

And lastly, the "monday paradox". In the version where she knows, in advance, she's going to be told it's monday, then the .5,.25,.25 space is clearly not correct because there's exactly 0 chance it's ever tuesday, so conditioning and coming up with 2:1 isn't valid BECAUSE THIS IS NOW A DIFFERENT EXPERIMENT THAT NEEDS A DIFFERENT SPACE. And obviously making a p,p,.5(1-2p),.5(1-2p) behaves perfectly. On the other hand, when she's told in advance that she will always be told what day it is, then:

Upon waking up (heads = 1/2, monday = 3/4, tuesday =1/4)
Upon being told monday (heads 2/3, tails 1/3)
Upon being told tuesday (heads 0, tails 1)

3/4*2/3 = 3/4*1/3+1/4*1 = .5

And it all works.

You should be able to embed the two experiments into the same space, actually, if the experiment goes flip, tell monday rules, go to sleep

a=it is before I sleep, before I've been told the rules, and the coin is heads
b=it is before I sleep, before I've been told the rules, and the coin is tails
c= it is before I sleep, I will always be told the day, and the coin is heads
d= it is before I sleep, I will always be told the day, and the coin is tails
e= it is before I sleep, I will always be told monday, and the coin is heads
f= it is before I sleep, I will always be told monday, and the coin is tails
g= it is monday, the rules are always-monday, and the coin is heads
h= it is monday, the rules are always-monday, and the coin is tails
i= it is monday, the rules are always-day, and the coin is heads
j= it is monday, the rules are always-day, and the coin is tails
k = it is tuesday, the rules are always-day, and the coin is tails

With the relations a=b, c=d, e=f, g=h, j=k, i=j+k, a+..+k=1, a,..,k>0, every expected property should be observed. I hope.

Consider the following sequence of related experiments.

Assume throughout that Beauty is not only given amnesia but also has no sense of time while awake. She has no sense of how long she's been awake since her last sleep. However, when awake within the experiment she will be aware that she is awake within the experiment.

All cases have the same settup where Beauty is told Sunday Night before she goes to sleep that a fair coin will be flipped after she goes to sleep.

Also assume the experimenters have technology which unobtrusively allows them to read Beauty's mind at any time and accurately tell them what credence she holds at that moment in the proposition that the coin fell heads. Beauty cannot tell when her mind is being read.

Beauty is told all this to begin with.


1.) Beauty is also told that in either case, Heads or Tails, she will be awakened in the morning and put back to sleep after an hour until the following Sunday at which time the experiment is over. What credence should Beauty have for Heads when she finds herself awake within the experiment?

Answer: 50%
Both cases H,T are identical.



2.) Same as 1) except Beauty is told her mind will be read once if Heads, and twice if Tails. If Tails her mind is read 1 minute after waking and again 31 minutes after her awakening. What credence should Beauty have for Heads when she finds herself awake within the experiment?

Answer: 50%
She reasons that the situation is identical to 1) where she clearly has credence 50%. They can read her mind as many times as they want, it doesn't change her credence that Heads was flipped. Her experience hasn't changed.



3.) Same as 2) except Beauty is told that if they read her mind and her credence is 50% they will take that as an indication she would be happy to bet on Heads and so they will place a $100 bet for her on Heads. What credence should Beauty have for Heads now when she finds herself awake within the experiment?

Answer: 50%
Whether they read her mind or place bets for her, nothing has changed from 1) to affect her credence the coin landed heads. Her experience remains identical to 1). However, when told of the bet she replies, "Well, if you are going to place two bets for me on heads when you know it's Tails I want at least 2-1 odds on the bet. Otherwise, play tricks with your own money."



4.) Same as 3) except Beauty is told that if Tails, 30 minutes after awakening she will be instantly put to sleep and reawakened 1 millisecond later with amnesia and no sense of how long she has been awake. To keep her total awake time the same when Tails, they will put her to sleep again 1 hour and 1 millisecond after her first awakening. What credence should Beauty have for Heads now when she finds herself awake within the experiment?

Answer: 50%
Beauty recognizes that while her experience is no longer identical to 1) it remains indistinguishable to her. If there were no mind reading nor bets she would certainly not change her credence because of a 1 millisecond loss of consciousness in the middle of the Tails awakening - all else being equal. And the mind reading and bets remain just as irrelevant as before.



5.) Same as 4) except Beauty is told it will be a 1 hour instead of 1 millisecond loss of consciousness in the middle of her Hour Long Tail Awake Time. What credence should Beauty now have for Heads when she finds herself awake within the experiment?

Answer: 50%
Beauty recognizes her experience will still be indistinguishable from 1). However they chop up her 1 hour Tails awake time with amnesia, having no sense of time her subjective experience of it remains the same. Chopping the time up has no bearing on which way the coin landed.



6.) Same as 4) and 5) except Beauty is told it will be a 24 hour loss of conciousness in the middle of her Hour Long Tail Awake Time. So half her Tail Awake Time will be Monday morning and the other half on Tuesday morning. What credence should Beauty now have for Heads when she finds herself awake within the experiment?

Answer: 50%
Like she reasoned in 5), they can chop up her Hour Tails Awake Time however they like, it has no bearing on either her subjective experience of it nor on whether the coin landed heads or tails.



2'.) Recall in 2) there was no interuption of consciousness, just the mind readings. In 2') it's same as 2) except the experimenters tell Beauty they're going to just ask her what her credence is for Heads rather than read her mind. To do this they give her a short term memory loss drug so that not only does she have no sense of time but when Tails and asked the second time she has no memory of having been asked the first time. What now Beauty's credence for heads?

answer: 50%
Beauty reasons that having her credence checked by asking rather than mind reading has no bearing on how the coin landed. She already knew in 2) that her credence would be getting checked twice when Tails. If the Tails double-checking didn't change her credence when by mind reading it shouldn't change it doing it the old fashioned way by asking Her.

If she changes her credence when asked because she thinks the act of being asked is twice as likely when Tails, then she must hold this alternate credence for the entire Hour Awake time whether she is being asked at the moment or not. So it would not be the "being asked" that would change her credence but knowing she was going to be getting double-asked when Tails. But she knew just as well in 2) that her credence was being double-checked when Tails by mind reading. If knowing the double check of her credence by mind reading didn't change her credence then knowing the double-check by asking shouldn't either.

So she maintains 50% whether her credence is checked by mind reading or by asking. Of course if they introduce betting in a 3') she will still demand at least 2-1 odds to bet on heads when asked her credence.



6'.) Same as 6) except they inform Beauty she will be asked her credence 1 minute after each awakening rather than reading her mind at those times. Her credence in heads?

Answer: 50%
Same argument as 2') together with argument in 6).



Notice 6') is the original Sleeping Beauty problem.

PairTheBoard

Probabilities and frequencies are not the same thing. They are connected via the law of large numbers, which requires independence. If this experiment is repeated many times, and A_n is the event that the coin is heads on Sleeping Beauty's n-th awakening, then the events A₁, A₂, A₃, ... are not independent. In the absence of independence, it is possible for the underlying probabilities to differ from the generated frequencies.

I am really not understanding how it is a contradiction that two observers with different info get different odds.

When I'm dealt two cards it's .5% to be aces from a villains view, and when I look at one card and see an ace it's 1/17 from mine.

Is this issue somehow more paradoxical?

This is a great answer

Here's my shocked face

This is misleading. She receives information that she already knew she was going to receive, so it's technically not additional information.

However, before she was woken up, she would have told you, "When I wake up, I know the odds will be 2/3 from my perspective".

That's the 1/2 argument from wiki.

It is correct she receives no new evidence when woken up. She receives it before being woken up.

She knows before she wakes up, there are 3 possible universes she can wake up in. MH MT TT. These 3 universes will exist with equal possibility.

What is the flaw in this argument?

I'm sure TomCowley will jump on this if I don't say anything.

I deleted my first post in this thread (right before Tom's post).

I read 3 different versions of this problem and as such my explanation didn't match up with the OP.

I stand by the thesis of my post that 1/3 is absolutely the correct answer, and I am not phased by the fact that some very intelligent people disagree with me.

+1. I never really got why the answer isn't totally obvious. Of course the coin flip is 50-50 not matter what and of course if sleeping beauty has to bet every time she is woken up it is obvious what she should bet on. Like if you woke her up 1000 times on tails and once on heads she should obv pick tails and be willing to lay odds.

"These 3 universes will exist with equal possibility."

That's not really an argument, that's just asserting the answer. Also, MT and TT are the same universe.

If you're interested in seeing where your argument might break down (or at the least is fuzzy) then check out this article by Nick Bostrom. I haven't read the whole paper yet but he claims to have a position that is different from both the 1/3 view and the 1/2 view. He explains each view in detail and points out the (arguably) problematic steps in each argument.

Fwiw I currently side with the 1/3 view. Bostrom seems to imply that the step in the argument that requires an "appeal to intuition" is problematic, but I have yet to see why it is.

The problem seems like an extension of this problem: She is put to sleep and only awoken after tails is thrown. She knows this, then her answer to her credence as to which side the coin landed will be 100% tails. So, if you're awoken 2/3 of the time when tails was thrown, you should believe tails was thrown 2/3 of the time.

I'm sure this is a stupid question, but I'm genuinely curious.

How is SSA not flawed in the same way as this:

"AA vs 72o is 50/50. You either win or you don't."

I can understand how SSA is viable with incomplete information, but in the OP we have complete information of the scenario.

This question is very different than the OP.

This question is extremely misleading.

"When she awakes on Monday..."

You are giving me the information that it's Monday. Does she have it?

What happens if the coin falls head. She never wakes up again and presumably dies?

If SB assumes no matter what she will awaken, she should assume 1/2 every time she wakes up without knowing the day of the week.

If SB assumes she dies if the coin lands heads, then she's more likely to wake up in the other world, therefore 1/3 is the answer.



As I've already said, the SSA approach only becomes viable with imperfect information, and in the OP ITT, we have perfect information. In the Nick Bostrum article, there is no perfect information.

These are vastly different questions.

I think the wiki on SSA is missing the clarification "Note that "randomly selected" is weighted by the probability of the observers existing" that's on the SIA page and letting "all other things equal" do the work without comment. Otherwise, yeah, wow, it's awful. I don't claim to be any kind of authority on SIA/SSA though.