This thread is going in circles and many people seem to be talking past one another.

TomCowley got it right when he said that the whole "paradox" is based on the confusing definition of probability (or lack there of).

There is no problem if one asks a concrete question, such as "What odds should she require to break even if she bets on it being Monday whenever she wakes up" or "If she guesses that the coin was flipped either heads or tails upon waking up, how often is she right (in the limit that the experiment is repeated many times" or anything like that. There is no paradox there, and we can all get the answer.

Things only get confusing when you say, "What is the probability of ..." and then people get confused as to whether they are talking about what odds she should lay or what is her subjective belief about the coin's result, etc.

I would argue that "probability" is just a middle-man definition that can be used to answer a concrete question, so why get caught up with it's definition. Instead, just answer the concrete question properly and then conclude you understand the problem.

It does exactly follow. I do not need to make up new experiments to prove my point. She can afford to invest more on tails for a given payout then she can on heads. That is the point. She has greater credence in the proposition that the coin was tails then she does in the proposition that the coin was heads. The problem is solved. There is no paradox.

If you cannot show that she wins the same by calling "tails" or "heads" when asked the question in this experiment, then it is not 50/50. Postulating new experiments or variations is just an attempt to cloud the issue.

Just to be clear, the experiment you described is different than the OP. By making the balls unique you have now said that she does not get asked the question twice. That is the case where she is put to sleep, awakened on heads on Monday and awakened on tails on either Monday or Tuesday, not both Monday and Tuesday. You can only get 50/50 if you somehow change the experiment to not count one of the opportunities to answer the question for tails. Then you have removed the bias that skews the answer from the 50/50 inherent in the coin. In the betting scenario that would be the case when she only wins once if she guesses tails both times that she wakes up. But that is a different experiment.

The poster above was right that this thread was going in circles but still I will attempt one more circle.

It doesn't follow as my 1st experiment shows. It would follow only if every bet is independent of any other bet. It isn't because being in MT and TT are not independent events. Your betting argument collapses here.

In first experiment I described (the one with numbered balls) she is betting two times if she draws black ball no 1.

Anyway, TomCowley and PTB argued the case better than me and Jason1990 showed exactly where 1/3'ers need to apply indifference principle and where 1/2'er do it.
I am happy to go few more circles here as I myself was changing sides a lot before arriving at what I now believe is correct resolution of this paradox. I just ask you to show will to understand what 1/2'ers are getting at and choose arguments you use for your stance wisely (ie. don't choose arguments which applied to different scenario yield unacceptable conclusions).

I'm afraid this doesn't work as you intend. As jason1990 often points out here, all probabilities are conditional. The 1/3,1/3,1/3 model is intended to describe the probability model for what Beauty's rational credence should be when she finds herself awake within the experiment. So in your (3) the naked probability P(H) is the naked probabality in a probability model which as a model is conditioned on the circumstance that Beauty is awake within the experiment. ZJ is right about this one.


PairTheBoard

This argument is at its root not about philosophy or probability, it is semantics. It is about the meaning of "What is her credence in".

The wager format removes that uncertainty. If she wagers each time she is asked the question, then clearly she wagers twice if it is tails and once if it is heads so choosing tails is the 2/3 favorite. If the experiment is configured so that there is only one wager on heads and one on tails, then it is 50/50, but that is the case where one of the times she is awakened on tails doesn't "count". Then it is a trivial case which is not worth discussing.

If you cannot see this, then fine. I am done with this.

I think it's clear in this case that he has gained information when he sees he's in the green room. It's not at all clear that Beauty has gained information finding herself awake. In fact, there's no proof she does. So the analogy is not equivalent on that point. Furthermore, you are not arguing your Sleeping Beauty 1/3 case based on her Times Awake within the experiment. After all, you objected to my (1)-(6) mind reading sequence where I specifically kept her total Time Awake per experiment constant. Your argument in SB is based on the number of times Her credences are checked. So your analogy here fails on that point as well.

PairTheBoard

Before she woke up, it could have been tuesday with heads flipped, the fact that she woke up within the experiment means it wasn't. This is information.

Originally Posted by PairTheBoard
"Guessing heads, She's right 1/2 the time "per experiment run" and 1/3 the time "per awakening". Which is the correct way to verify the validity of her credence?"


This is what the OP says:
That does not define "the correct way to verify the validity of her credence." It simply tells us that she may have a credence and if so we know what it is upon any awakening. If she is rational, with no other information her credence should be consistent throughout the experiment. The OP certainly doesn't define how we should check its validity. That's the "problem". If the OP had defined the answer to that for us there would be nothing to discuss.

I reiterate. To my mind, the correct way to check the validity of a credence for the outcome of a coin flip is to check it per coin flip.

PairTheBoard

Reading this thread, I've changed my opinion from "1/2" (initial intuition based on no information..) to "1/3" (frequencies and gambling) to "wtf?" (TC's long posts).

Me too but switch the order of 1/3 and 1/2 :P

While I can see problems with both sides, and im definitely not as confident as some on here that its correct, im firmly in the 1/3 camp because of the above. If you cant accept 1-1 odds, how can it possibly be 50%? That is just flat contradictory imo.

Because the bet is circularly defined. The number of times you place the bet is dependent on the outcome of the bet. You can't apply normal EV equations here.

The way the bet was defined in punter's post doesn't even really relate to the SB problem anyway, it is just saying you have a 50% chance of winning one unit, 25% chance of losing one, and 25% chance of losing two.

Anyway this thread seems to have just degenerated into people coming up with dozens of different analogies and complaining that not every single one has been responded to. It's getting dumber every post.

This. I have more sympathy for the 1/3 school partially because I think of probability in terms of what would be required to have a 0EV wager. If we can't define it by that, or by the results of a large number of trials, then how do we define it?

I show you a 1x1x1 box and tell you there's a cube inside (sitting on a face, edges parallel to the box edges). I tell you that if the side length is .5 or less, the cube is red (and not red otherwise). I tell you that if the face area is .5 or greater, the cube is blue (and not blue otherwise). What is the probability the cube is red? blue?

Ya this is settled as 1/2 for me

Explain to me how you settle on 1/2?

If you awaken her under the format of the experiment and ask her to pick heads or tails with a $1 prize for the correct answer, what is her EV for a heads pick or a tails pick?

That is easy to answer if you consider multiple runs. After 1000 experiments she will be awakened 1500 times, with 500 heads and 1000 tails on average. Thus the total award is $500 if she guessed only heads and $1000 for only tails. Her EV is $0.33 for heads and $0.67 for tails.

Since EV is (prob. of event) x (profit from the event), the probability must be 1/3 head and 2/3 tails.

I truly cannot understand how you can see this very simple argument and say 1/2. Please explain that to me. Show me where the error in this is.

It's meaningless nonsense. Try answering the cube question.

Is the answer "we can't know" because we don't know what process led you to choosing the cube? But how is this relevant to a situation where the process is defined by a coin flip?

It has nothing to do with the OP. If he could not follow the EV post he is just confused.

There is no error in the EV calc, but the answer is still 1/2.

Under normal situations placing bets and expected value are directly related. In this situation it isn't : the odds are still as stated (1/2) except placing a bet is circular. You only win the bet (the EV shows 2:1 for tails) because you get to bet twice when right. But the actual odds are different.

I will give you the study I did that finally convinced me: Lets say that there is a big red button in the center of the table. It can only be pressed once. If she presses it and hasnt pressed it before, she gets $1 if it is tails, loses $1 if it is heads. If she presses it and has pressed it before, (or if she never presses it,) nothing happens. You might think of this button as an indicator of whether tails is a favorite*. If tails was really 2/3, she would press this button every time, since her EV would then be $0.333..., right?

You can compute from simple algebra (I've omitted the details) that her optimal strategy is to press the button 1/2 of the time. The reason is basically she is trying to maximize the number of times she presses the button when it is not monday. And her EV under the optimal strategy is $0.125. Maybe this can help bring light on the idea that any betting in this situation is not indicative of the probability, but rather just a trick around being able to bet multiple times.

--


(*its not, but under your line of thinking, it would be)

Sorry, but this is just incorrect.

The OP is a normal situation.

This is the abnormal situation that decouples the economics of EV from the statistics. The button causes tails to not payout on the second press. If the coin is tails and she presses on Monday then she gets the $1 payout. If she then presses again on Tuesday the coin is tails but the button cancels the payout so the EV is decoupled from the statistics of the coin/wakeup probability.

I would have called finding the optimum a very simple calculus problem but I agree that the answer is .5 and that the EV is $0.125/wakeup, but as I have already shown this is fundamentally different from the OP.

If tails is truly 2/3 then why do you need to bet twice to achieve +EV by betting it? Isn't it better to say that tails is 50% but you are betting tails twice?

Anyways I am very convinced, if the above didnt convince you I donno what to say.

Edit: I feel you basically understand the problem as well but we are just arguing the semantics.

I agree that the coin is 50/50. It is the combination of the coin and the system for waking SB up that creates the 2/3. The betting was just a way of quantifying her credence in the proposition. If she does not bet when she wakes up, then the money no longer tests the credence accurately.

The extreme case of waking her up once on tails and never on heads illustrates the effect. When she wakes and is asked her credence for heads, she says 0. Again, the coin is still 50/50 but the combination of the coin and the system for waking her in that case makes the probability of tails 1.

I also am quite convinced, so I guess we will have to agree to disagree.

Ok first, the shoot in her sleep argument doesnt hold because in that game, waking up proves it isn't heads, meanwhile in this game when you wake up it can still be heads.

The crux of the question is, do you get any new information when you wake up? And the answer has to be no. Because there is no way to distinguish anything.

This is my best argument yet for the halfer side and I have not seen this in any paper I have read or forum discussion etc. This argument I believe is very good.

Again, lets suppose there are 3 different interview rooms labelled HM TM TT. Lets suppose that the thirder view is correct:

When you are in room HM or TT, you are asked and you say that it is 1/3 heads. Then the interviewer says the experiment is over, and asks you. You say 1/2 heads correctly. Theres no contradiction yet.

So during the interview, you think it is heads 1/3. And you know that 2/3 of the time, the interviewer will say that the experiment is over, and 1/3 of the time, it wont be and youll just fall asleep and not remember. So 2/3 of the time, it is 1/2 heads. That means 1/3 of the time it is 0% heads. (2/3)(1/2) + (1/3)(0) = 1/3. Checks out, okay, still no contradiction. Note that halfers would say that 1/2 = (2/3)(1/2) + (1/3)(1/2) -- doesnt matter when you ask them, they give the same answer. Also note that even though you will be asked in the future and answer 1/3, you reasoning now in the first interview will reason that in the next interview the true chance of heads is 0, since obviously if the experiment isn't over before your next awakening, it must be tails.

Okay, so now lets change the rules of the game. On Sunday, the experimenter flips the coin, but on Tails he flips it again and if it is tails again, he just shoots her in her sleep. Halfers would now argue that the credence of heads is 2/3, while thirders will argue 1/2, for the exact same arguments as before. (You can check this yourself.)

So again we use the above argument. Thirders would claim that 3/4 of the time, the interviewer will say that the experiment is over, and 1/4 of the time you will fall asleep. So 3/4 of the time, it is 1/2 heads, and 1/4 of the time, it is 0 as before. But (3/4)(1/2) + (1/4)(0) = 3/8 which isnt 1/2. Contradiction.

The bolded can't both be true. In the latter, 3/4 of the time it's 2/3 heads (HHM, HTM or THT), so (3/4)(2/3) + (1/4)(0) = 1/2. (I realise the experimenter doesn't actually flip again when he flips heads, but it doesn't matter if he does and it makes it clear that HM is more likely than TT).