Ralph got *no* new information, and yet his optimal plan has changed. Which plan should he follow???

Note on Mixed Strategies:

  (1/3)*0 + (2/3)*(1/3)*4 + (2/3)*(2/3)*1 = 4/3.

But if Ralph adopts this plan, then when he arrives at some unknown exit he will surmise that (on the assumption that he is actually following his own plan) the chance that this is the first exit is 3/5, the chance that it is the second exit is 2/5. The expected utility of exiting is thus 8/5, which is higher than the utility of his original plan. Should he therefore decide to exit?

-Jamie

This problem is due to Ariel Rubenstein and Michele Piccione, and is discussed formally in their GAMES AND ECONOMIC BEHAVIOR 20 (1997), 3-24. Any arithmetic errors in this posting are certainly mine and not theirs. Any conceptual confusions are also extremely likely to be mine and not theirs.

What credence should she state in answer to our question?

-Jamie

p.s. Don't worry, we will awaken Beauty afterward and she'll suffer no ill effects.

p.p.s. This puzzle/problem is, as far as I know, due to a graduate student at MIT. Unfortunately I don't know his name (I do know it's a man). The problem apparently arose out of some consideration of the Case of the Absentminded Driver.

p.p.p.s. Once again, I have no very confident 'solution' of my own; I will eventually post the author's solution, but I am not entirely happy with that one either.

Poor? Work all of the time and barely make ends meat? Feel like ending it all? Come to me first!

At my clinic, we will buy you a lottery ticket. Now everyone knows that your odds of winning our state lottery are only 1 in a million, but the $20 million dollar jackpot this time can almost certainly be yours if you just call me now! When you arrive you will be given a warm bath and fed a four star meal of your choice. When you are done being pampered, you just lie down and go to sleep. While you are sleeping, our trained personnel will watch the TV to see if your lottery ticket is a winner. If it isn't, then you just pay our small $250 service fee and go on about your day. If it is, then you will be repeatedly awakened into lavish surroundings and by beautiful women every day for the rest of your life. How you ask?!? Well, at the end of every 30 minute interval, we will wipe your memory clean up to the time you arrived! The mathematics behind this miracle are complicated, so I won't bore you, but if we suppose that you will live 50 more years, then your odds of awakening to find that you have won are amazingly over 50%! Wouldn't you pay $250 to have a 50% chance of winning the lottery?!? The number is 1800-IMAFOOL. Operators are standing by.

Sometime in the future we devise a procedure for cloning objects by replicating their atomic structure. We are even able to clone humans. One day, Andy shows up at a clinic and makes arrangements to be cloned. He has an unusual request. He asks that after he is put under anesthesia, a coin be flipped and that he only be cloned if it should come up tails. Furthermore, he asks that before he is woken he be placed randomly into one of two identical rooms. His clone, if one exists, should be placed into the other room.

When Andy awakens, he realizes that he doesn't know whether or not he has been cloned or even if he is the clone. Soon, before he learns anything about the toss or whether he was cloned, his friend Bob shows up. Bob knows all about what Andy was going to do, and that he is in one of the two identical rooms. He randomly picks one and goes in, to find Andy there in bed. After shooting the breeze for a bit, they have the following conversation:

Andy: "Before I was put under anesthesia, I asked myself 'what is probability that I will ever wake up due to heads?'. Since it would happen if and only if the coin flipped heads, I knew then the odds of that were 1/2. Now I have woken just like I knew I would, and I still know nothing about the coin toss result. The odds of heads must still be 1/2."

Bob: "Wait. I know more than that. I picked a door at random and walked in. If you were cloned, I would have found you for sure. If you weren't I would have had only a 50% chance of finding you. Since the event that I would find you was twice as likely if the coin landed tails, the probability now that the coin did actually land tails is twice that of heads given that I actually did find you. That is, heads has probability 1/3."

Andy: "No. Look at it from my point of view. When you randomly picked a room, there were only two possibilities for me: either you would find me or you wouldn't. Since you picked at random, not only are these possibilities equally likely, but the event that you would find me is completely independent of the coin toss. Hence, the conditional probability of heads given that you found me must be the same as I would have assessed it before you came - 1/2."

After arguing for a bit, they are unable to reach an agreement on the probabilities. They finally settle on a wager. Bob will give Andy $4 if the coin is heads, and Andy will give Bob $3 dollars if it is tails. Now each man has exactly the same information as the other, and yet each man has calculates this wager to be favorable for himself. Obviously it can't be favorable for both of them – which man has reasoned incorrectly?

>Please resolve the following paradox:
>
>Beauty has a large purse full of money. She hasn't counted the
>money in a long and really has no idea how much she has. Before
>she goes into the experiment, her friend Sally tells her that
>each and every time she wakes up Sally will offer the following
>bet.
> If the coin was tails, then Beauty pays Sally 2 dollar.
> If the coin was heads, then Sally pays Beauty 5 dollars.
>Sally says she will offer it every time, even if she already knows
>she will lose from the previous time. When beauty wakes up she
>thinks the odds that the coin was tails are 2/3. Now, if the coin
>was heads then I will win 5 dollars. On the other hand, if the
>coin was tails then I will pay her 2 dollars. But wait, since I am a
>rational person, I will end up paying her twice! In reality, I am
>going to end up paying her 4 dollars if the coin came up tails.
>Working out the math, Beauty estimates that her expected loss if she
>takes the bet is 4*2/3 - 5*1/3 = 1 dollar. Clearly she should have
>accepted, what happened?

This is the most clearly perplexing formulation of the problem that I have seen yet.

SB can think
"There are three equiprobable cases.
(1.) The coin was heads, and I have an option of winning $5.
(2.) The coin was tails, this is my first awakening, and I have an option of losing $4.
(3.) The coin was tails, this is my first awakening, and I have an option of losing $4.
Therefore I reject the bet."
As you say, this is clearly wrong.

Here is another version of it. This one is not confused by memory loss, lack of consciousness, or non-conservation of persons.

You are a loyal agent of the baron. You know that all of his agents are as intelligent as you are. You are suddenly summoned to the king's court, where a courtier tells you

"The king decided to play a game. He tossed a coin. If it came up heads, he put the gift of five manors in an envelope, and had a representative of your baron summoned, to be offered the envelope. If it came up tails, he put two separate documents, each ordering the confiscation of two manors, in two separate envelopes, and had two representatives of your baron simultaneously and separately summoned, each to be offered an envelope."

"You are such a representative. Here is an envelope. Do you want it?"

The answer is obviously "accept the envelope". Yet the logic which I have used to defend "1/3" for the Sleeping Beauty problem says otherwise.

> I'm not convinced by it either. In fact I retract what I said earlier
> about 1/2 being as good an answer as 1/3 due to an "ambiguity" in the
> problem. I now think the question in the original problem can most
> reasonably be taken as to estimate the conditional probability of
> heads *given* the knowledge that Beauty has when she wakes up on
> Monday or Tuesday.

This conditional probability bit is a sham. She could have conditioned on "I will wake up on Monday or Tuesday" before she went to sleep. People seem to have more confidence in the Beauty problem than they do in my cloning problem. I don't see the difference, so I am going to go ahead and rephrase the cloning problem that I just posted in terms of Beauty. Consider this:

Beauty is rushed to the hospital after experiencing a sharp pain in her head. When she arrives, doctors administer preliminary tests which are very pessimistic indeed. They tell her that she will live at least 23 more hours. Then, there is a 1 in 2 chance that she will drop dead in the next hour. However, they tell her that if she does live past 24 hours, then all danger from this condition will have passed. Beauty is horrified at the news. Her daughter is touring Europe, and would most likely only be able to make it back to see her mother for a short few minutes before she dies. Nevertheless, steps are taken to bring her daughter back. Seeing her daughter before she dies is the most important thing on Beauty's mind.

Over the next hour, the doctors perform more extensive tests which conclusively determine if Beauty will be dead at the end of the day. Beauty, a mathematician, doesn't want to know. Instead, she wants to engage in the same procedure that we have seen repeatedly on this thread. She is to be put to sleep. If the test says she will die, then she is awakened once. She isn't told of the test results, and is just left to live out the rest of her day. If the test says she will live, then for the next ten years Beauty will have her memory cleared every other day. (She will get to experience two days, each time, before having her memory wiped).

Beauty is put to sleep. When she wakens she feels great piece of mind knowing that her odds of dying by the end of the day are now less than 1 in 1750. Wow she thinks, I really don't have much to worry about with that whole disease thing. This procedure, while not actually having done anything to decrease her odds of dying, has put her mind at rest. The day seems to pass quickly, just like any other day, with her mind at ease. She is told that in 15 minutes, she will be possibly be in the hour period where she risks death. Beauty laughs as she thinks about how improbable that is. "Oh wait, my daughter should be here by now", thinks Beauty. "Oh well, I'm tired. I'll just see her in the morning." And with that thought, Beauty turns out her light and goes to sleep.

Wouldn't you have done the same thing?

[...]

> Everyone allowed a decision is acting rationally, but the one forced to
> play (the third party / the new clone) is getting screwed over. I don't
> see a paradox here.

True, but this isn't where the paradox lives. The paradox (to me) lies in this situation:

A coin is flipped 10 times in a row.
If (and only if) it comes up tails each time, then you are cloned a million times.

When you are wakened, after the possible cloning, you (and clones of you) are asked "Were they all tails?"
If you answer correctly, you live happily.
If you answer incorrectly, you are tortured and killed.

The paradox is that thirders would answer yes to this question. (I wouldn't - but can't justify my decision)

You wake in a cell with total amnesia.
On a paper in front of you is the following:

Yesterday, Beauty and Betty were placed in cells just like this one. A coin was tossed. If it landed heads, then Beauty alone was given a memory wiping drug. If it landed tails, then both Beauty and Betty were drugged. A copy of this note was placed in both of their cells. You are either Betty or Beauty.

What are the odds that you are Beauty?
What are the odds that Betty got drugged?
If you learn that you are Beauty, what are the odds that Betty was drugged?

After a time, Beauty landed near a daisy. What credence did she then give to the statement "The coin came up heads"?

You were present when a crime was committed, but you took no part in it. You have been arrested, and are tried for this crime.

After you have given your evidence, the judge sentences you (this country has an unusual judicial system). Her sentence is "You will now be taken from this court and locked in a prison cell. Meanwhile the trial will continue and we will hear the evidence of the witnesses. If we find you not guilty, you will be released from prison at 10 p.m. tomorrow – this short sentence will be for your failure to prevent or report the crime. But if we find you guilty, you will be visited at 10 p.m. tomorrow by someone who will administer a soporific and 24-hour memory-wipe. This treatment will repeated until you have passed 100 nights in prison, and then you will be released at 10 p.m. Psychologists advise us that this will have a reformatory effect on you."

You know that you are innocent of the crime. But there is a chance that the witnesses will lie and the court will reach an unjust verdict – you estimate this chance as 20%.

Some time later, you wake up in prison. How do you rate your chance of being released the same evening?

As a thirder, I know that the answer is 5/104. But my gut feeling is that it is .802 (ok, my guts aren't that precise - let's say somewhere around 0.8). I am finding it hard to reconcile this at present.

You visit Dr. Evil's lab for an experiment. The following protocol is explained to you and you remember it at all times.

On Sunday you're put to sleep with drugs. That night, Geddy Lee flips a fair coin and rolls a fair die.

On Monday and Tuesday, according to the conditions below, you may be woken up by Nurse Chapel. After this, Will Smith pulls out his penlight and pushes the history eraser button, and you forget being woken up.

On Monday, you're woken up if and only if the die shows 1.

On Tuesday, you're woken up if and only if: the coin shows heads and the die shows 1; or, the coin shows tails and the die shows 2.

On Wednesday, Dr. Franklin shoots you up and sends you on your way.

...

You're woken up by Nurse Chapel.

What's the chance the coin is heads?

The thirder answer is "1/3". I would like to know how halfers answer.

My main qualm with this line of argument is that it relies on the sequence of results and the pure SB experiment is a single-shot. But then again, my gut feeling for the definition of probability relies on repetitions of trials and I think that probabilities are ambiguous for experiments that by definiton cannot be repeated.

Upon awakening, Beauty cannot remember which kind of ticket she got. Does she hope that she got an A ticket, or a B ticket, or is she indifferent between them?

What does she answer, if Tim asks?

When Beauty awakes, her experience is the same regardless of whether the coin is heads or tails. Therefore, her awakening cannot provide any new information about whether the coin came up heads or tails. Since her credence is 1/2 before the experiment, it must be 1/2 during the experiment.

[Since I am not a halfer, I must disagree with this somewhere. My disagreement starts at the word "therefore."]

This is similar to Jim Ferry's initial response to the original Beauty posting. Jamie Dreier has also posted in an apparently sophisticated way in favor of halfism, but I never really understood his argument.

And finally: there is a version of halfism which is not usually made explicit by halfers which says that all probability statements (including Beauty's statements of credence) must refer to the randomness *in the coin* only. Using this probability space, the only three possibilities for credence are 0, 1/2, and 1, so the answer is clearly 1/2. This position appears odd, but it is actually in accord with much of modern statistical practice.

Mike

Here's one argument:

Given no information about the coin toss, the probability is 1/2 by definition. In order for this probability to change, sleeping beauty has to be given information that distinguishes the cases in any way. However, due to the parameters of the problem, she cannot distinguish the heads case from the tails case by simply being woken up. Therefore, she does not have any information as to the result of the coin toss, so the probability she must state is 1/2.

(while sleeping beauty being woken multiple time may allow others to gain information from the waking, her memory loss does not allow her to do so)

Aaron

It's one thing to say that later I will know the answer, and another to say that later I will know that the answer is 1/2. If she knows that later she will know the answer is 1/2, then she knows it is 1/2 now.

Tell me what you think.

Bill Vanyo

This is effectively the same as SB's scenario.

> Sorry, I missed most of the Sleeping Beaty thread, and most of the stuff I
> did catch seemed a bit complicated without the foundations. Can anyone
> point me in the right direction for decent, clear arguments for both the
> halfist and thirdist positions? (I've already seen the questions on Nick
> Wedd's page, what I want now is some answers to check my ideas against)

Here is something I posted a while back, explaining the (IMHO) three most convincing arguments for 1/3. Attached at the end are some details for the objective advisor setup. [From the postings of mine on Nick's page, you could probably tell that I was a halfer. I am now a thirder.]

Conditionalization Argument
Summary: After waking, if Beauty learns that the day is Monday, then she should think P[heads]=1/2. (There is a simple argument for this based on the fact that knowledge of the day was exactly the piece of information she lost by the drug, and she thought P[heads] was 1/2 before she lost that information). On the other hand, if she learns that the day is Tuesday, then she should obviously think P[heads]=0. If she doesn't know the day, then the probability must lie somewhere in-between.

Objective Advisor Argument
Summary: We can arrange for Beauty to meet with an objective advisor to discuss the probability of the coin toss. We can do it in such a way that the advisors presence can't influence Beauty's assessment of the coin toss, while the advisor, through Beauty being awake, does gain information supporting a particular outcome of the toss. Through standard probability techniques the advisor will compute that the probability of heads is only 1/3, and advise Beauty as such. If the situation is setup correctly, this can be the case even if Beauty and the advisor disclose every piece of information they have to each other. *For the record, all attempts to get Beauty to meet with an advisor who thinks that P[heads]=1/2 have resulted in Beauty getting new information. (and that information always allows her to agree with P[heads]=1/2)

New Information Argument
Summary: The situation can be modified so that Beauty does seem to receive new information. Instead of waking her more times when the coin lands tails, instead wake her the same number of times regardless of the toss outcome, always without memory of any other wakening. Clearly, by symmetry, she should think that heads and tails are just as likely upon waking. So, suppose that we wait five minutes and then put her back to sleep unless the coin landed tails or it is her first day up. When she finds that she has not been put back to sleep she has new information which leads her to P[heads]=1/3. Further, the cases in which she is awake after five minutes are exactly the cases in the original problem.

-- Details for the Objective Advisor argument --

One day, Sleeping Beauty visits the Institute of Memory Loss to be part of an experiment. She will be put to sleep on Sunday night, and a coin will be tossed. If the coin lands heads, then Beauty is woken either on Monday or Tuesday at random. On tails, she is woken on both Monday and Tuesday, subject to usual memory loss.

In this experiment, Dr. Blue and Dr. Green, the world's foremost authorities on statistics, are hired to assist Beauty in her probability reasoning. If the coin landed heads, then one of the Statisticians is chosen randomly to help assist her. If the coin landed tails, then one of the PhDs gets her on one day, at random, and the other gets her on the other day.

Now, using this setup, lets analyze the situation from Beauty's perspective in a random wakening. Clearly, after she wakes up and before she finds out which doctor she will visit, her probability assessments will be just as they are in the original Sleeping Beauty situation. Now, at that point, she knows that she will either meet with Dr. Blue or Dr.Green, but doesn't know which. Can her probability assessment change when she finds out which will advise her? Clearly not. Both men are, by symmetry, equivalent to her. One is not any more associated with heads or tails than the other. Hence, her probability assessment of the coin toss should be the same when she visits her advisor as it was when she woke.

So, what will her advisor say about the probability of heads? Well, if the coin had landed heads, then he would only have had a 50% chance of meeting Beauty. On the other side of the coin, his meeting her would have been a certainty. Since he does get to meet her, he will believe that the probability of heads is only 1/3, when Beauty and him discuss the probability of heads in the coin toss, he will be quite insistent about this (you know how these Ph.D. types can be). As it stands, it is possible that the Statistician has been able to more accurately assess the situation with the piece of information that he has which she doesn't – he knows the day.

Well, lets have him tell Beauty the day. How can this affect her probability assessment? Again, in this variant, it can't. By symmetry, neither Monday nor Tuesday is more indicative of heads or tails than the other. After this disclosure, Beauty and her advisor have exactly the same relevant information. It stands to reason that if they don't agree on the probabilities at this point, then one of them is wrong.

Suppose we didn't erase Beauty's memory, and she knows this. Everything else remains the same as in the previous problem -- on heads, she gets wakened on Monday; on tails, she gets wakened on Monday and Tuesday.

Obviously, when we ask her the question on a wakening, she will either think:

1. "Well, I don't remember being wakened, so it must be Monday. This means that the probability of the coin being heads was 1/2."
2. "Well, I remember being awakened yesterday, so today is Tuesday and the probability of the coin being heads is 0."

The halfer, then, thinks that Sleeping Beauty's knowledge of the erasure is irrelevant, while the thirder seems to think it is.

Assume that it is one half. Repeat the experiment a million times. On each awakening, Beauty bets one dollar, even money, that the coin was tails. If the probability is one half then she'll break even in the long run. But she won't break even in the long run. On each heads she'll bet once and lose. On each tails she'll bet twice and win twice. In the long run she'll be about a half million dollars ahead. So the assumption that the probability is one half is not correct.

So in our new problem Sleeping Beauty concludes that it was a tail 2/3 of the time, and a head 1/3 of the time, whatever she is told about the day. Now if we step back to the modified problem, she wakes up but is not told the day. She thinks to herself... "If I had been told the day, I would give the answer 1/3 WHATEVER I was told. Here I haven't been told, but should that bother me?"

Even if you transform the original question into one where, in the case of tails, she is put to sleep again each day and asked again each day repeated indefinitely, the outcome would not change. This is because on Wednesday the answer would be 1/4, on Thursday it would 1/5, and so on and so on. Each of those days would have a unique answer while 1/2 would be true in both outcomes on the original monday. Meaning that responding with 1/2 would have 2/(x +1) odds of being correct while 1/3 and every other possible answer would only have 1/(x+1) odds where x equals the number of days that the experiment is allowed to continue. This means that responding with 1/2 would always give Sleeping Beauty the greatest odds of being correct.

The ambiguity is also exposed by our inability to agree on a definitive test for SB. While some suggest replacing the question 'what is your credence...?' with a bet on every awakening, others offer to provide her with a ticket that wins $5 on tails and ask her how much the ticket is worth on every awakening. In any normal situation, every reasonable test would lead to the same result (and the problem doesn't remain unsolved for years...). We need a very unique setting involving memory-erasing drugs to distort our intuition and leave such an ambiguity unnoticed.

Appendix – The Absentminded Motorist Paradox

This paradox is stated above.

In fact, AM is a recursive reasoning paradox (my term). It creates an infinite reasoning loop that doesn't seem to reach a definite solution. It is similar to Newcomb's paradox where you try to make a decision after an all-knowing being tried to predict it.

To understand how AM is similar we simply need to go further with AM's reasoning: After he decides he should exit, he must immediately rethink this decision considering that he knows himself to reason in that way: he is inclined to enter at the first exit, so this can't be the second exit (he never reaches it). But now, if this is the first exit then he should drive on to the next exit – and we're back to square one.

It is clear that the problem here is different and cannot be solved for different reasons than the SB paradox.