The Sleeping Beauty Problem: Any halfers here?

[Stephen Tashi]

But you are mixing up the probability of heads in one way of framing the calculation, with her credence, which can be shown to be quite different.
.

Before we get into the details, let me clarify my viewpoint. My viewpoint is that the posterior probability of heads P(heads | SB is awakened) cannot be calculated from the information in the problem. Asking for it's value is an ill-posed problem and is irrelevant for planning a betting strategy. As you pointed out, planning a betting strategy should use P(Heads = 1/2)., not P(Heads|wake). The value P(heads)= 1/2 is used regardless of whether one is a "halfer" or "thirder" or agnostic about P(heads|SB awakened).

In my view, when SB answers "What is your credence that the coin landed heads?", she gives a dishonest answer if she reports the number based on her betting strategy. However, many posters in this thread are willing to accept her report as her credence that the coin landed heads, so they don't object to it. Her report is dishonest because she is not considering a simple bet "You get $1 if the coin lands heads". Instead, when planning her strategy, she considering that paying a price X for the bet obligates her to pay that price every time the bet is offered. Two different versions of the bet might offered. On Monday, it is an even bet. On Tuesday (if offered) the bet is a sure loser.

Since calculating P(Heads| awake) is an ill-posed problem, SB cannot offer a "credence" for the event (Heads | awake) unless she makes some assumptions. She does not need to calculate P(Heads| awake) unless the experimenter is stickler and can propose a bet on "You get $1 if the coin lands heads" that is a "pure" bet - i.e. a bet with no conditions that she might have to buy another bet with different expected payoff. (I myself haven't been able to formulate a "pure" bet. that could be offered to SB during the experiment.)

From my reading, the general opinion of those who have studied the SB problem is that one cannot distinguish between the "halfer" and "thirder" positions by bets that can be offered during the experiment, provided we assume SB is rational and plans a betting strategy that is independent of her opinion of P(Heads|awake).

Both "thirders" and "halfers" are incorrect to assert that their answer for P(Heads|awake) is the unique correct answer. Both the "thirder" and the "halfers" are correct that that one can create a probability model that is consistent with their answer for P(Heads|awake) and does not contradict the information given in th SB problem. The computation of a betting strategy is independent of such a probability model.

 

[Stephen Tashi]

So it depends on SB's knowledge of what betting opportunities will be made available to her.

Most posters in this thread ( including myself) assume that when she is awakened SB knows the how the experiment is conducted and that she will be asked about her credence for "the coin landed heads" every time she is awakened - but you're right, the Wikipedia statement of the problem doesn't make this crystal clear. It does say that the amnesia drug makes her forget "her previous awakening". It doesn't say she forgets other facts.

 

[Ken G]

Before we get into the details, let me clarify my viewpoint. My viewpoint is that the posterior probability of heads P(heads | SB is awakened) cannot be calculated from the information in the problem. Asking for it's value is an ill-posed problem and is irrelevant for planning a betting strategy. As you pointed out, planning a betting strategy should use P(Heads = 1/2)., not P(Heads|wake). The vale P(heads)= 1/2 is used regardless of whether one is a "halfer" or "thirder" or agnostic about P(heads|SB awakened).

Yes, it seems that the whole concept of "probability" is creating problems, but the correct betting odds should be clear enough-- 1/3 heads is the break-even betting strategy, and that's all "credence" means.

In my view, when SB answers "What is your credence that the coin landed heads?", she gives a dishonest answer if she reports the number based on her betting strategy.

Let us simply define credence by her fair betting strategy, that's really all the concept is supposed to entail.

On Monday, it is an even bet. On Tuesday (if offered) the bet is a sure loser.

Yet that is always true. If you are playing bridge, and you analyze a finesse as having a 50-50 chance, of course in one deal, it's a sure bet, and another, it's a sure loser. But that's just not what credence means, credence is not some kind of "actual probability", it is simply the correct betting strategy given the information you have. It makes no difference if she takes the bet every time or not, the correct betting strategy is always 1/3 heads because she never gets any additional information in one trial versus another. You don't have to finesse every time you get a certain hand in bridge-- yet the odds are still 50-50 if that's all you know. I suspect the problem in this discussion is the halfers are seeking some kind of "actual probability" of heads, and indeed we both just did a calculation using P(heads)=1/2, but that isn't what credence is-- credence is simply the fair bet, any time the bet is made given no information beyond what is supplied in the experiment.

From my reading, the general opinion of those who have studied the SB problem is that one cannot distinguish between the "halfer" and "thirder" positions by bets that can be offered during the experiment, provided we assume SB is rational and plans a betting strategy that is independent of her opinion of P(Heads|awake).

This is what is false. There is clearly a single break-even betting strategy in the experiment as described, and it is clearly 1/3 heads. The 99 day version makes this very clear-- you already agreed the fair bet in that experiment is X=1/50 for "today is Monday", so how can the fair bet be 1/2 heads when heads can only pay off on Monday?

 

[Ken G]

only needs to know that, in order to reach what conclusion?

In order to reach the conclusions that the fair bet is 1/3 "the coin was heads," and the fair bet is 2/3 "today is Monday."

If the conclusion is to be that she is willing to pay 33c for a bet that pays $1 if the coin landed Heads and zero otherwise, she needs more information than that - specifically, what are the probabilities of me being offered the same bet on every other day.

But the problem does clearly specify this. Using the Wiki version:
"Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends."

So there it is, SB is interviewed every time, so her credence is about a bet she is given every time. It's clear as a bell, no additional assumptions needed. Of course the experimenter cannot couple the offering of the bet to the outcome of the coin, that would clearly be cheating and would do violence with the entire concept of credence. If credence were dependent on the uncertain and dishonest wiles of the experimenter, then you are saying the odds of winning any game of chance (the details of poker are of course irrelevant) cannot be known unless you know you are not being cheated. That goes without saying, it is simply what "odds" mean. If you assess the chances of a sports team winning, you never say "assuming none of the players are throwing the game intentionally," because that also goes without saying in the concept of odds.

 

[Stephen Tashi]

you already agreed the fair bet in that experiment is X=1/50 for "today is Monday", so how can the fair bet be 1/2 heads when heads can only pay off on Monday?

I'm not sure which experiment you're talking about. Which post was it?

 

[andrewkirk]

The experiment already stipulates that SB is given the same bet every time, that's what credence means

Not according to the SEP definition. There is no mention of 'every time' in that definition, because it only envisages one bet. This situation has the potential for two bets, where the number of times the bet is taken is correlated with the result of the bet. That is a completely different situation from the one envisaged in the SEP definition, which implicitly assumes that there is only one bet. Hence there is room for multiple different interpretations as to how the single-bet SEP definition might be generalised to apply to this case.

For convenience, here is the SEP definition again:

Your degree of belief [credence] in E is p iff p units of utility is the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E.

 

[stevendaryl]

The error in the Thirder position is to assume that the three probabilities are the same. There is no mathematical rule to justify that.

I would say that's a conclusion, not an assumption. If you repeat the experiment many, many times, for many, many weeks, then, letting N be the number of weeks, then on the average:

• There will be N/2 times when Sleeping Beauty is awake and it's Monday and it's heads.
• There will be N/2 times when SB is awake and it's Monday and it's tails.
• There will be N/2 times when SB is awake and it's Tuesday and it's tails.
• There will be N/2 times when SB is asleep and it's Tuesday and it's heads.

So the three possibilities: (Monday, Heads), (Monday, Tails), (Tuesday, Tails) occur equally frequently. So if Sleeping Beauty's credence for each possibility is the same as the relative frequency of that possibility in repeated experiments, then she has to give equal likelihood to those three possibilities.

That's the relative frequency argument: the relative frequency of heads among the events in which she is awake is 1/3.

 

[stevendaryl]

To me, the halfer position has a very implausible consequence. There are three possibilities given that Sleeping Beauty is awake:

1. (Heads, Monday)
2. (Tails, Monday)
3. (Tails, Tuesday)

The subjective probabilities of the three, given that she is awake, and given that today is either Monday or Tuesday, must equal 1. So

P(H & Monday | Awake) + P(T & Monday | Awake) + P(T & Tuesday | Awake) = 1

So if P(H & Monday | Awake) = 1/2, and P(T & Tuesday | Awake) > 0, then it follows that

P(H & Monday | Awake) > P(T & Monday | Awake)

This in turn implies:

P(H & Monday) > P(T & Monday)

which implies:

P(H | Monday) > P(T | Monday)

This is what's bizarre to me about the halfer position. They want to say that P(H | Awake) = P(H), because being awake doesn't tell you anything new about whether it's heads or not. But the consequence of that position is that P(H | Monday) > P(T | Monday). If being awake tells you nothing about whether it's heads or tails, why would you say that it being Monday tells you something? How can the fact that it's Monday make heads more likely than tails?

What's especially bizarre about this is that, as far as the thought experiment goes, it doesn't make any difference whether you flip the coin on Monday morning or on Tuesday morning, since it's not necessary to consult the result until Tuesday morning. So in the case where the coin flip is on Tuesday morning, the halfer position implies that:

• If Sleeping Beauty is told that today is Monday, and that tomorrow morning, a coin will be flipped, she will say that it is more likely that the result will be heads than tails.

That's completely bizarre.

 

[Dale]

we need to know whether she knows, before the experiment begins, that she will be offered a bet every day

The bet is implied by the definition of credence. So it is every time she is asked about her credence, which is every interview.

So on that definition, and with those very specific rules about betting, (which were not stated in the original problem statement)

They were stated in the original problem. Beauty is asked about her credence in each interview so the implied bet is necessarily offered each interview also.

 

[stevendaryl]

Not according to the SEP definition. There is no mention of 'every time' in that definition, because it only envisages one bet. This situation has the potential for two bets, where the number of times the bet is taken is correlated with the result of the bet. That is a completely different situation from the one envisaged in the SEP definition, which implicitly assumes that there is only one bet. Hence there is room for multiple different interpretations as to how the single-bet SEP definition might be generalised to apply to this case.

I'm not so sure that it matters. It is true that the number of opportunities to bet depends on the coin result. However, each individual opportunity to bet seems to fit the definition.

 

[Buzz Bloom]

The probability of being woken on Monday in the Tails situation is 1/2 the probability of Tails, which is 0.5 times 0.5 = 0.25. The same goes for Tuesday-Tails.

Hi:andrew:

You seem to be making the same mistake that I made before I recignized my mistake as I explain in my post # 498.
There are two awakenings for tails, not one.

Regards,
Buzz

 

[Dale]

posterior probability of heads P(heads | SB is awakened) cannot be calculated from the information in the problem.

I have shown that this is false back in post 255. At this point, I would ask you to stop repeating this false claim.

she gives a dishonest answer if she reports the number based on her betting strategy. ...The computation of a betting strategy is independent of such a probability model.

And this is a mistake on your part. The honest price that she would buy or sell the bet is the value of her credence. If she answered with a value that she would not bet on then she would be dishonestly representing her credence

 

[Ken G]

I'm not sure which experiment you're talking about. Which post was it?

Take the correct calculation of X you did in post 549 and apply it to the version of the puzzle with 99 days. You should get X=1/50, meaning, SB would take 49 to 1 odds against it being Monday.

 

[Ken G]

Not according to the SEP definition. There is no mention of 'every time' in that definition, because it only envisages one bet.

The puzzle stipulates that she is interviewed every time. That means she is given the same interview every time, because to assume otherwise is to assume dishonest collusion on the part of the experimenters. If they give her a different interview to try to fool her into making a bad bet, they are cheating. It goes without saying in any puzzle that the experimenters are not intentionally cheating.

This situation has the potential for two bets, where the number of times the bet is taken is correlated with the result of the bet.

There is a bet in every interview, that's what establishing the credence, in the interview, means, according to the definition you gave and I am also using. I see that Dale is making that same point.

 

[.Scott]

The sleeping beauty problem is a well known problem in probability theory, see e.g.
https://en.wikipedia.org/wiki/Sleeping_Beauty_problem
http://allendowney.blogspot.hr/2015/06/the-sleeping-beauty-problem.html
https://www.quantamagazine.org/solution-sleeping-beautys-dilemma-20160129
or just google.

I believe we need to examine the judgement and perhaps the intentions of these experimenters more closely.
Did these experimenters dupe Sleeping Beauty into taking a dangerous cocktail of drugs in the name of science?

It is the amnesia drug in particular that is of most concern. These drugs, such as Valium and high doses of alcohol are potentially addictive and are associated with suicide. Assuming these experiments are of sufficient value to science (and I am sure they are), they should be designed for minimal use of these drugs.

However, it is clear from the methodology that no such consideration was given.
Why was amnesia induced after Sleeping Beauty responded to the questions on Tuesday?
When the coin came up heads, why was amnesia induced at all?
In fact, once the last interview was conducted on either Monday or Tuesday, why were any additional drugs used at all?

I will also note that this experiment was designed in 2000, decades after the over-prescription of Valium had been widely publicized.

 

[Ken G]

All humor aside, that is actually a good point-- there's no need for amnesia drugs for a heads flip. The answer to the puzzle, 1/3 credence for heads, is the same as long as SB knows she would have forgotten Monday's events had tails been flipped, nothing else is needed. The puzzle specifically says she only forgets the awakening, so the drugs are only needed after the Monday interview and only if tails came up (Returning to humor vein, at least the way it is set up allows the experiment to be done double-blind, though of course the interviewers will also need to be given the amnesia drug, further complicating the ethical dilemma.)

 

[Stephen Tashi]

I'm not so sure that it matters. It is true that the number of opportunities to bet depends on the coin result. However, each individual opportunity to bet seems to fit the definition.

How can an answer that Sleeping Beauty obtains without considering the probability of ( heads| awake) indicate her credence of that event?

The bet (heads'awak) on Monday is even odds. If offered, the bet on (heads | awake) on Tuesday is sure loser. Just because Sleeping Beauty doesn't know which bet she is making doesn't mean they are "the same to her". Her calculations for what to bet don't involve computing the probability of (heads | awake). She simply uses the fact that p(heads|) = 1/2 and accounts for the probability that she may be for to make two different bets. If she was treating the bets as being the same, her calculations would show they had the same pay-off.

 

[Stephen Tashi]

Take the correct calculation of X you did in post 549 and apply it to the version of the puzzle with 99 days. You should get X=1/50, meaning, SB would take 49 to 1 odds against it being Monday.

SB is assume to be rational. A rational "halfer" or a "thirder" doesn't use their estimate of p(heads | awake) to decide on their bets. What SB bets should be iindependent of p(heads|awake).

 

[Stephen Tashi]

I have shown that this is false back in post 255. At this point, I would ask you to stop repeating this false claim.

Request denied. The claim isn't false. Your two-line derrivation of the "thirder" claim is not a valid proof. If the "thirder" answer was unique , the "halfer" answer would not be another solution.

And this is a mistake on your part. The honest price that she would buy or sell the bet is the value of her credence. If she answered with a value that she would not bet on then she would be dishonestly representing her credence

SB computes her answer without computing P(heads | awake). In fact, her answer is computed using p(heads) = 1/2. So how does her answer indicate her credence for the event p(heads | awake)?

A rational "halfer" gives the same answer a "thirder" does, so how does credence in P(heads | awake) have anything to do with the bet?

 

[stevendaryl]

How can an answer that Sleeping Beauty obtains without considering the probability of ( heads| awake) indicate her credence of that event?

I don't understand the question. You wake Sleeping Beauty up. In case she doesn't remember, you remind her of the rules. Then you ask her if she wants to bet on whether it's heads or not. Of course, she will consider the probability of heads given that she's awake in deciding which way to bet.

 

[Stephen Tashi]

I don't understand the question. You wake Sleeping Beauty up. In case she doesn't remember, you remind her of the rules. Then you ask her if she wants to bet on whether it's heads or not. Of course, she will consider the probability of heads given that she's awake in deciding which way to bet.

No.

A rational Sleeping Beauty will know that if she always answer X then the expected net price she must pay for doing so is: (1/2)X + (1/2)(2X) = (3/2)X. Her expected gain from giving that answer is (1/2)(1) + (1/2)(0). So she solves (3/2)X = (1/2) to obtain X = 1/3. This calculation is done using P(Heads) = 1/2 and does not involve computing p(Heads | awake).

 

[stevendaryl]

No.

A rational Sleeping Beauty will know that if she always answer X then the expected net price she must pay for doing so is: (1/2)X + (1/2)(2X) = (3/2)X. Her expected gain from giving that answer is (1/2)(1) + (1/2)(0). So she solves (3/2)X = (1/2) to obtain X = 1/3. This calculation is done using P(Heads) = 1/2 and does not involve computing p(Heads | awake).

I don't understand the distinction you're making. I would say that that calculation IS the calculation of P(H | Awake). Sleeping Beauty knows that if the experiment were repeated many times, then approximately 1/3 of the times in which she is awake will be when the coin toss was heads, and 2/3 of the times in which she is awake will be when the coin toss was tails. If you want credence to line up with relative frequency, then the credence of heads, given that she is awake, should be 1/3.

 

[Ken G]

SB is assume to be rational. A rational "halfer" or a "thirder" doesn't use their estimate of p(heads | awake) to decide on their bets. What SB bets should be iindependent of p(heads|awake).

Please take your post 549 and use it to calculate, via the very method you used there, the X for it being Monday in the 99 day version. You should get X=1/50, just do the same thing you did in post 549. Is that indeed what you get?

The bet (heads'awak) on Monday is even odds. If offered, the bet on (heads | awake) on Tuesday is sure loser. Just because Sleeping Beauty doesn't know which bet she is making doesn't mean they are "the same to her". Her calculations for what to bet don't involve computing the probability of (heads | awake). She simply uses the fact that p(heads|) = 1/2 and accounts for the probability that she may be for to make two different bets. If she was treating the bets as being the same, her calculations would show they had the same pay-off.

There are two different ways SB can arrive at her credence that the day is Monday, one involving P(heads)=1/2 from the start, and another involving P(heads/awake)=1/3. (That second calculation asserts that her credence that it is Monday equals P(heads|awake) + P(tails|awake)*1/2) That both give the same answer shows that the correct credence that it is Monday is 2/3, which you got using the first calculation involving P(heads) rather than the second calculation involving P(heads|awake). But note that P(heads)=1/2 is not her credence that the coin is heads, P(heads/awake)=1/3 is. This is the mistake you are making.

 

[Stephen Tashi]

I don't understand the distinction you're making. I would say that that calculation IS the calculation of P(H | Awake).

SB would make the same calculation if she's a "halfer"

Consider this experiment:: A coin is flipped. A red hat is put on your head.. If the coin lands heads, you will be asked "What is a fair price for the the bet that you get $1 if the coin landed heads?" You give some answer X and pay that amount to purchase the bet. If the coin landed tails, you are required to purchase the bet again at the same price.

Is the answer X that you choose to give equal to your creedence for the event "The probability that (the coin landed heads | given I'm wearing a red hat)?

 

[Ken G]

The whole discussion comes down to this. Given the rules of the scenario, where SB is interviewed every time she is wakened (not just some arbitrary set chosen by cheating experimenters), we may conclude that every time she is interviewed, it is true that P(heads)= 1/2 and P(heads|awake)=1/3. That's the halfers and thirders right there. The thirders are correctly responding to the question asked, which is, what is her credence given that she has been wakened. The halfers are answering the wrong question, they are simply asking what is the probability that the coin came up heads in the first place, which is not a probability that can be altered.

I think it becomes clear what the halfers are thinking if we consider a 99 day version, but we make the following change. If heads are flipped, SB is interviewed only on Monday. If tails, SB is interviewed on a single randomly chosen day, sampled equally from the 98 days after Monday. In that case, her credence that it is heads is clearly 1/2, that's the halfer thinking. The halfers are claiming her credence is not affected if we change the experiment to interview her all 99 days if it's tails. That's clearly wrong, but to make it perfectly clear, we can simply look at her credence that it is Monday in the two situations. The halfer thinks her credence it is Monday is 1/2 in both versions of the 99-day experiment, because if the coin flips a heads, she gets interviewed on Monday. But that's not half the times she is awakened.

 

[Stephen Tashi]

To me, the halfer position has a very implausible consequence. There are three possibilities given that Sleeping Beauty is awake:

1. (Heads, Monday)
2. (Tails, Monday)
3. (Tails, Tuesday)

The subjective probabilities of the three, given that she is awake, and given that today is either Monday or Tuesday, must equal 1. So

But the consequence of that position is that P(H | Monday) > P(T | Monday). If being awake tells you nothing about whether it's heads or tails, why would you say that it being Monday tells you something? How can the fact that it's Monday make heads more likely than tails?

"Halfer" answers will seem bizare if you don't think in terms of proability model that is consistent with "halfer" answer.

Compute the probabilities using the "halfer" probability model that I've mentioned before. One probability model consistent with "halfer" viewpoint describes how to select from the 3 events you mentioned in the following manner:

1) Flip the coin

2) If the coin is heads, (Heads & Monday) is selected. (This represents picking the only day SB is awake in the experiment when the coin lands heads.)

If the coin is tails, pick one of the events (Tails&Monday) , (Tails&Tuesday), giving each a probability of 1/2 of being the event selected. (This represents picking one of the events in the experiment "at random" given the coin has landed tails.)
In words, one can visualize SB's thought process upon awakening as: "I might be in either version the experiment the heads-version or the tails- version. I'll assume there is an equal chance of being in either. Given that I'm in the heads-version, it must be Monday. Given I'm in the tails version, it could be Monday or Tuesday. I'll assign an equal probability to those events."
One can object that SB is making unwarranted assumptions, but so is a "thirder" probability model. One can object that SB gets the "wrong" answers, but they are not wrong by the "halfer" probability model. The are "wrong" if the "thirder" model is assumed.

What's especially bizarre about this is that, as far as the thought experiment goes, it doesn't make any difference whether you flip the coin on Monday morning or on Tuesday morning, since it's not necessary to consult the result until Tuesday morning.

SB knows nothing definite about the day or whether the coin must have been flipped already. If you're thinking about the mental processes of someone who knows the coin has not been flipped, you're not thinking about SB's mental processes.

 

[bahamagreen]

I think Ken G #585 has got it.

After being briefed on the experiment and before being put to sleep Sunday evening rational SB wonders to herself:

"What is my present credence of heads right now, and what will be my credence of heads when I find myself future awake in an interview?"

What are thirders figuring would be her answers? If the answers are different, how is imagining being awake later different from waiting to find herself awake later, as far as calculations?

 

[stevendaryl]

"Halfer" answers will seem bizare if you don't think in terms of probability model that is consistent with "halfer" answer.

Forget about probability models. Just consider that Sleeping Beauty has just awakened. You tell her that today is Monday and that tomorrow we're going to flip a coin to decide whether to wake her up (and if you do wake her up, she'll have no memory of Monday having happened). You ask her: What's the likelihood that the coin flip tomorrow will be heads. The halfer answer has to be 2/3.

What probability model can make that sensible? How does something that we're going to do tomorrow after the coin flip affect the probability of the coin flip?

 

[stevendaryl]

I think Ken G #585 has got it.

After being briefed on the experiment and before being put to sleep Sunday evening rational SB wonders to herself:

"What is my present credence of heads right now, and what will be my credence of heads when I find myself future awake in an interview?"

What are thirders figuring would be her answers? If the answers are different, how is imagining being awake later different from waiting to find herself awake later, as far as calculations?

Suppose instead of waking once if heads and twice if tails, we said that you wake zero times if heads and two times if tails. Then it would make perfect sense for Sleeping Beauty to say:

"The probability that the coin flip will be heads is 1/2. But if on Monday or Tuesday I'm awake, I'll know that it was definitely tails."

So the number of times being awakened definitely affects her answer of "What credence of heads will you give when you're awakened?"

 

[Stephen Tashi]

The thirders are correctly responding to the question asked, which is, what is her credence given that she has been wakened.

One does not need to be "thirder" to compute what answer should be given.

Consider this situation. A coin is flipped. The bet is "You get $1 if the coin lands heads and you lose $2 if the coin lands tails". Is the fair price for the bet equal to your credence for the event "The coin landed heads"?

No, it isn't.. By the definition of credence cited several times in this thread, your credence for "The coin landed heads" is supposed to be the fair price you set for the bet "You get $1 if the coin lands heads" - with no other consequences.

 

[Stephen Tashi]

Forget about probability models.

I don't think questions about probability can be decided by forgetting about probability models - but I understand the frustration they bring.

Just consider that Sleeping Beauty has just awakened. You tell her that today is Monday and that tomorrow we're going to flip a coin to decide whether to wake her up (and if you do wake her up, she'll have no memory of Monday having happened). You ask her: What's the likelihood that the coin flip tomorrow will be heads. The halfer answer has to be 2/3.

What probability model can make that sensible?

I just gave you one.

How does something that we're going to do tomorrow after the coin flip affect the probability of the coin flip?

If the coin is flipped before SB is awakened for the first time, how do "thirders" explain that SB's awakening goes back in time and affects how the coin landed?

Conditional probabilities do not imply any physical cause-and-effect relations. Neither "halfers' nor "thirders" should expect to justify their answers by some cause-and-effect physical model.

 

[Ken G]

One does not need to be "thirder" to compute what answer should be given.

Consider this situation. A coin is flipped. The bet is "You get $1 if the coin lands heads and you lose $2 if the coin lands tails". Is the fair price for the bet equal to your credence for the event "The coin landed heads"?

Of course not, and that's why I am not using that meaning. I am saying, the credence is the fair price she would pay, X, to receive a $1 payoff if she's right. That's exactly what is 1/3 for "heads" each time she is wakened, as is easy to show in repeated trials.

In my opinion it is crucial to avoid probability arguments, because then people ask questions like "what is the actual probability the coin came out heads," and of course there is nothing that goes back in time and changes that in some absolute sense, like a force on the coin. We should instead think in terms of fair betting odds, which is what credence actually is. One can do it with probability, but subtle issues enter, like what counts as information that can cause a reassessment of a probability. But betting odds make the situation way easier, it becomes an actual way to make or lose money. We can actually do the experiment, and bilk the halfers out of their life savings.

 

[Ken G]

Suppose instead of waking once if heads and twice if tails, we said that you wake zero times if heads and two times if tails. Then it would make perfect sense for Sleeping Beauty to say:

"The probability that the coin flip will be heads is 1/2. But if on Monday or Tuesday I'm awake, I'll know that it was definitely tails."

So the number of times being awakened definitely affects her answer of "What credence of heads will you give when you're awakened?"

Yes, this example shows clearly that being wakened does indeed involve new information that changes SB's assessment of the heads probability. What is confusing the halfers is they think the heads probability is a set thing, specified when the coin is flipped, but probabilities actually mean what is consistent with the information you have. In the way the puzzle is formulated, it's subtle what that new information is, so your version makes it clear that being awakened is a form of information. For halfers reading this, if you play bridge, consider that every bridge hand has a probability of being dealt, but much of the skill of bridge amounts to updating that probability using information gathered during the bidding and play. So "the probability they have the queen" is not set by the deal, because probabilities are more active animals than that. That this is a subtle point is my reason for avoiding probabilities in favor of simply odds payoffs, ergo the relevance of the game of poker for those who have played it.

 

[Stephen Tashi]

Of course not, and that's why I am not using that meaning. I am saying, the credence is the fair price she would pay, X, to receive a $1 payoff if she's right.

But you are adding the condition that she has to pay X twice if she's wrong. So you are making the bet have consequences similar to the example I gave.

That's exactly what is 1/3 for "heads" each time she is wakened, as is easy to show in repeated trials.

I agree that saying 1/3 is the correct strategy. What I'm saying is the definition of credence for an event E is supposed to be a "pure" bet on E. Your opinion is that because the experimenter uses the words "What is your credence for the event 'the coin landed heads'" that SB is being offered a "pure" bet on that event. She is not. She is being offered the bet: " If coin lands heads you win $1 and if it lands tails you lose twice what you offered for the bet".

 

[Ken G]

But you are adding the condition that she has to pay X twice if she's wrong.

No, I'm saying that she has to pay X every time she takes a bet that is wrong. That's just how betting works.

I agree that saying 1/3 is the correct strategy.

This is the crucial point-- that's all credence means. So you are not doing anything wrong in your mathematics, and you cannot be made to lose money. You are simply not using the definition of credence correctly-- that definition is, you pay X, and lose X, every time your bet is wrong, regardless of how often that may be. If X is the ratio of the cost of a bet to the payoff, and if you break even with some given X, then X is your credence. It's not an abstract definition, it's a practical one, relevant to all betting games of chance.