This month’s puzzle featured two variations on the famous Sleeping Beauty problem. We compared the original problem to a famous visual illusion, the Necker cube, which is a two-dimensional figure that can be perceived as a three-dimensional object in one of two different orientations, with both perceptions equally valid. The Sleeping Beauty problem has spawned two clear-cut camps known as the halfers and the thirders. There are dozens, perhaps hundreds of papers offering highly sophisticated arguments supporting one side or the other. It is astonishing and a little unsettling to some that such a situation can arise in a branch of mathematics, in this case probability or decision theory. Are there ambiguities at the core of these disciplines that can undermine them? Certainly not in this instance: The underlying specific procedures here are very well-defined and do yield answers that everyone can agree on when problems are specific enough. Yet there is plenty to argue about in the Sleeping Beauty problem, so let’s dive into the original scenario.

The famous fairy-tale princess Sleeping Beauty participates in an experiment that starts on Sunday. She is told that she will be put to sleep, and while she is asleep a fair coin will be tossed that will determine how the experiment will proceed. If the coin comes up heads, she will be awakened on Monday, interviewed, and put back to sleep, but she won’t remember this awakening. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday, again without remembering either awakening. In either case, the experiment ends when she is awakened on Wednesday without being interviewed.

Whenever Sleeping Beauty is awakened and interviewed, she won’t know which day it is or whether she has been awakened before. During each awakening, she is asked: “What is your degree of certainty* that the coin landed heads?” What should her answer be?

*(This is sometimes worded as “credence,” “degree of belief,” “subjective probability” or “subjective certainty.”)

As I discussed when I presented the puzzle, the halfers, whose position was first articulated by David Lewis in 2001, assert that the answer is one-half. Since the coin was fair, the chance that it came up heads is half. Sleeping Beauty receives no new information about the result of the coin toss when she is awakened. So her subjective probability that the coin came up heads should continue to be one-half.

Thirders, following and extending Adam Elga’s argument in 2000, argue that there are three possible situations in which Sleeping Beauty could have been awakened, which are indistinguishable to her. The coin could have come up heads and it is Monday, the coin could have come up tails and it is Monday, or the coin could have come up tails and it is Tuesday. Each of these is equally likely from her perspective, so the probability of each is one-third. Since the coin comes up heads in only one of these situations, her subjective probability that the coin came up heads is one-third.

One reason for the differences between the two camps is that halfers and thirders interpret the question in two different, though equally valid, ways, and translate it into slightly different mathematical problems. Credence is an intermediate mental construct that is defined differently by the two camps. As *Quanta* reader Josh put it, “Like most verbal paradoxes, [this dilemma] relies on underspecifying the precise question!” When the question is put in precise mathematical terms, there is no paradox, and both halfers and thirders get the same answer — just as the ambiguity of the Necker cube is resolved if the front face is specified, as in the accompanying illustration. There can be clear-cut answers, then, as we shall see in our well-specified variations — there’s no need to worry about the foundations of probability. It’s just a matter of being specific about what is asked, as we saw in an earlier Insights column.

And yet, in its original form, this problem has generated a huge philosophical and psychological chasm between the two positions, in a way that the Necker cube has not, since most people can flip quite easily from one percept to the other. Each Sleeping Beauty camp is passionate about its stance. I know from personal experience that arguments with someone in the opposite camp are usually futile. It’s far more productive to try and nail down the differences in thought between the two camps and get at least an intellectual, if not intuitive, appreciation of the opposing camp’s mindset. I have attempted to depict the differences in the thought processes experienced by the two camps in the chart below.

Halfers take an experimenter’s view: For them, the question is about which of the two arms of the study Sleeping Beauty is in — the heads (red) or the tails (blue) arm as depicted in the top part of the figure. Each of the arrows shown in the figure represents a completed “trial.” The detailed setup of the experiment — the amnesia, and the number of awakenings — is irrelevant, “a snare and a delusion” in the words of halfer Peter Winkler. In this view, Sleeping Beauty has no specific information as to whether the coin landed tails or heads because she is awakened per the original protocol regardless of what happens. So if the experiment were repeated many times, halfers would count the number of times she ended up in the heads arm of the experiment, relative to the total number of trials. Since it is given that the coin is fair, this will happen in one-half of the trials.

Thirders focus on the subject’s view: For them, the question has to do with which timeline she is more likely to be in when she is awakened, that of a heads or a tails toss. She is twice as likely to be in the tails timeline when she is awakened as she is to be in the heads timeline, as seen in the fact that there are twice as many arrows on the blue waking panels as there are on the red ones. The detailed setup of the experiment such as the amnesia and the difference in the number of awakenings in the two coin states — whether one, two, or a million — is now of vital importance and will drastically change the degree of belief the subject has about which coin-toss timeline she was awakened in. To a thirder, the information that Sleeping Beauty receives is contained in a combination of three things: 1) The details of the experiment protocol known to her beforehand, which in this case samples the two tosses differently, 2) the amnesia which makes all the awakenings identical from her point of view, and, crucially, 3) the realization that “I am now awake.” So if the experiment were repeated many times, thirders would count the number of awakenings that happen in the heads timeline relative to the total number of awakenings. Since there is only one awakening for a heads toss out of every three awakenings, the subjective probability of the coin toss having come up heads will be one-third.

To dig a little deeper into the halfer-thirder quarrel, at its heart it’s a disagreement about what constitutes valid information that Sleeping Beauty can use to update her probability about what happened in the coin toss. This is an extremely subtle point. Interested readers can check out a modified experiment I set forth in a comment dated January 27, 2016, which describes the kind of information that both parties agree compels Sleeping Beauty to update her probability for the coin toss per textbook Bayesian criteria. But this consensus fails in the case of the original Sleeping Beauty problem. I’ve come to believe that the gap cannot be bridged, and that the difference in approach perhaps arises from some deep intuitive source, such as gut-level responses arising from early training or kinds of mathematical problems encountered or even psychological or personality traits. This is not an unprecedented situation. The basis for another seemingly irreconcilable polarization — between liberals and conservatives — has been traced by the psychologist Jonathan Haidt to deep gut-level intuitions arising from emotional and personality differences. Are halfers better at visualizing points of view outside themselves? Do thirders tend to be temperamentally pragmatic? It might make for an interesting psychological research project.

OK, so we’ve seen that halfers and thirders completely disagree about nebulous things like credences and subjective probabilities. But here’s the key point: When challenged with a well-specified problem with clearly stated utility functions (probabilities of specific payoffs) linked to each alternative, skilled halfers and thirders have no difficulty in selecting the proper model to apply (even if it’s not their default one), applying the laws of probability correctly and getting the right answer! The proof of the pudding in probability problems is in the making of a concrete bet that has one answer that gives the best result, as in the two variations presented in this month’s puzzle. The halfers and thirders all got the correct answer, even though it was not their default stance. As a mathematics teacher or puzzle columnist would say, “Go figure!” On a more serious note, this shows that the laws of probability are indeed rock-solid, and the two dueling positions are only based on default gut-level points of view that do not get in the way of finding the right answer to a well-specified problem.

The two concrete variations we presented were based on a scenario created by *Quanta* reader eJ. They were as follows:

Variation 1:

Upon each awakening, Sleeping Beauty is presented with two bags of beans, marked “H” and “T.” She is instructed to reach into one bag, grab a single bean, and put it aside. At the end of the experiment, she will have to eat the bean or beans that she has pulled. She is told that the bags are filled with identical looking jellybeans (J) or poisoned pills (K), as follows:

- If the coin came up heads, bag H has 7J, and bag T has 7K.
- If the coin came up tails, bag H has 1J and 6K, while bag T has 6J and 1K.
You are Sleeping Beauty. Which bag would you pick, and what are your chances of survival?

Variation 2:

You, as Sleeping Beauty, are told that you have to go through the original experiment (without the beans) every week for many months, and the memory of each waking will be wiped from your memory. The evil chief scientist has determined that on your hundredth awakening in this series of experiments, you will be presented with the two bags of beans and instructed exactly as in Variation 1 above. If you pick a poisoned pill, you will die; otherwise, you will go free. Now which bag do you pick, and what are your chances of surviving?

In these two problems, you are given the same utility function, so the calculations are very similar. For each variation, you multiply the likelihood of the desired outcome by your probability estimate for the appropriate coin toss for every possible alternative and add all the cases to get your final likelihood of survival. Let’s check out how the halfer and the thirder would fare in the two problems. Remember, probability calculations boil down to nothing more than counting carefully. We discuss Variation 1 in detail here, and leave Variation 2 to the reader. The qualitative results for the two are identical.

If you are a halfer and you pick bag H:

- If the coin landed heads (your estimated probability: 1/2) you will definitely pick a J (1), giving 1/2 x 1 = 1/2.
- If the coin landed tails (
*p*= 1/2), you will need to be lucky enough to pick the jellybean out of the H bag on both Monday and Tuesday to survive, so your chances are (1/7 x 1/7) x 1/2, giving 1/98.

Adding the two probabilities makes your overall chance of survival 1/2 + 1/98 = 25/49 (51 percent).

If you are a halfer and you pick bag T:

- If the coin landed heads (
*p*= 1/2) you cannot pick a J (0), giving 1/2 x 0 = 0. - If the coin landed tails (
*p*= 1/2), you need to pick a J out of the T bag on both Monday and Tuesday to survive, so your chances are (6/7 x 6/7) x 1/2, giving 18/49.

Adding the two probabilities makes your overall chance of survival 0 + 18/49 = 18/49 (37 percent).

So if you are an inflexible halfer, you will pick bag H for both problems.

If you are a thirder and you pick bag H:

- If the coin landed heads (your estimate:
*p*= 1/3) you will definitely pick a J (1), giving 1/3 x 1 = 1/3. - If the coin landed tails (
*p*= 2/3), you need to pick the J out of the H bag on both Monday and Tuesday to survive, so your chances are (1/7 x 1/7) x 2/3, giving 2/147.

Adding the two probabilities makes your overall chance of survival 1/3 + 2/147 = 51/147 (35 percent).

If you are a thirder and you pick bag T:

- If the coin landed heads (
*p*= 1/3) you cannot pick a J (0), giving 1/3 x 0 = 0 - If the coin landed tails (
*p*= 2/3), you need to pick a J out of the T bag on both Monday and Tuesday to survive, so your chances are (6/7 x 6/7) x 2/3, giving 24/49.

Adding the two probabilities makes your overall chances of survival 0 + 24/49 = 24/49 (49 percent).

If you are an inflexible thirder, you will pick bag T for both problems.

Note the beauty of eJ’s scenario: Your choice of bag, H or T, perfectly reflects whether you are acting as a halfer or a thirder.

But wait, who is correct? We saw above that halfers and thirders model different problems. Which categories do these two variations belong to?

In Variation 1, your fate is decided after you have gone through one arm of the experiment or the other as in the top part of the diagram. This requires the kind of counting that is correctly modeled by a halfer. So that’s what you need to be. The correct bag to pick is H.

In Variation 2, your fate is based on a single selected awakening, picked, for all practical purposes, at random. This requires the kind of counting that is correctly modeled by a thirder, and so the correct bag to pick is T.

Our readers, whether they identified as halfers or thirders, picked the correct answers in both cases. So there you have it. Halfers are halfers, and thirders are thirders, and never the twain shall meet — unless they are solving a concrete problem and getting the same result! Here’s a quantum analogy: The Sleeping Beauty problem is like an unobservable particle in a quantum superposition of 50 percent H and 50 percent T among the ensemble of puzzle enthusiasts. As in quantum mechanics, when an actual measurement is made (a concrete question is posed), the state is correctly found by the ensemble to be either H or T.

I thank all readers for providing great insights, ideas and references. I’d especially like to thank Joel Pust for providing a link to the extensive philosophical literature, and Josh for citing and summarizing Berry Groisman’s paper declaring everyone to be winners. In awarding the Quanta T-shirt this week, I’d like to endorse the suggestion expressed to me by eJ: “The T-shirt winner needs to demonstrate some respect for both sides of the argument.” The T-shirt therefore goes to Josh, narrowly beating out the similarly excellent contributions of Paul Smaldino, Robert and Dan. I think everyone will agree that eJ deserves a T-shirt too, for proposing these two ingenious variations. Congratulations!

In the paragraph which begins

"Thirders focus on the subject’s view:"

…

"She is twice as likely to be in the tails timeline when she is awakened as she is to be in the heads timeline"

Unfortunately I would like to disagree with that.

Let's look at it from Sleeping Beauty's point of view.

When SB is awakened and interviewed there are three mutually exclusive cases:

Case 1. The coin toss was heads & the day is Monday

Case 2. The coin toss was tails & the day is Monday

Case 3. The coin toss was tails & the day is Tuesday

When she is awakened she doesn't know which case it is, but she can calculate the likelihood of each case using the definition of conditional probability.

(see https://en.wikipedia.org/wiki/Conditional_probability#Conditioning_on_an_event.)

P(B|A)=P(A&B)/P(A)

but apply it in this equivalent form:

P(A&B)=P(A)*P(B|A)

Case 1.

P(The coin toss was heads & the day is Monday) =

P(The coin toss was heads)*P(the day is Monday | The coin toss was heads)

P(The coin toss was heads) = .5

When the coin toss was heads and SB is awakened and interviewed it must be Monday.

P(the day is Monday | The coin toss was heads) =1.0

So the probability of Case 1. is .5*1.0 = .5

Case 2. (Case 3. is similar)

P(The coin toss was tails & the day is Monday) =

P(The coin toss was tails)*P(the day is Monday | The coin toss was tails)

P(The coin toss was tails) = .5

When the coin toss was tails and SB is awakened and interviewed she has no way of knowing whether it is Monday or Tuesday. Also, when the toss is tails, both days occur with the same frequency, so I suppose she would have to consider each day equally likely.

P(the day is Monday | The coin toss was tails) = .5

Hence the probability of Case 2. is .5*.5 = .25

Summarizing the three cases:

Case 1. P(The coin toss was heads & the day is Monday) = .5

Case 2. P(The coin toss was tails & the day is Monday) = .25

Case 3. P(The coin toss was tails & the day is Tuesday) = .25

So even though when she awakens there are two cases with tails, each of those only has a probability of .25 while the one case with heads has probability .5 .

What is her degree of certainty that the coin landed heads?

Her answer should be .5 .

Patrick,

From the experimenter's or God's eye view, your answer of 0.5 is absolutely correct.

But now think about it from SB's very narrow point of view. Suppose every time she wakes she says to her friend "I have a strong degree of certainty, in fact 2/3, that the coin landed tails, and only 1/3 that it landed heads. So I'll bet you a dollar that it indeed landed tails." If she does this everytime she is awakened in a series of experiments, her winnings will reflect the fact that that her belief was perfectly justified.

This is a subjective, pragmatic view of "degree of certainty." And it is equally justified. It's called the "Dutch Book" view and we use it in our lives every day.

I hope you can at least fleetingly, experience the way the other half thinks.

If I may elaborate even further, there's another point in your article where I think Sleeping Beauty should disagree with the "thirders":

"To a thirder, the information that Sleeping Beauty receives is contained in a combination of three things: 1) The details of the experiment protocol known to her beforehand, which in this case samples the two tosses differently, 2) the amnesia which makes all the awakenings identical from her point of view, and, crucially, 3) the realization that “I am now awake.” So if the experiment were repeated many times, thirders would count the number of awakenings that happen in the heads timeline relative to the total number of awakenings. Since there is only one awakening for a heads toss out of every three awakenings, the subjective probability of the coin toss having come up heads will be one-third."

I agree with 1,2&3. "the amnesia makes all the awakenings identical from her point of view" in terms of her ability to distinguish between the three cases based on what she actually experienced.

But that does not prevent SB from realizing that there are three different contexts in which she might have been awakened:

Case 1. The coin toss was heads & the day is Monday

Case 2. The coin toss was tails & the day is Monday

Case 3. The coin toss was tails & the day is Tuesday

When SB is awakened and interviewed, nothing stops her from using the definition of conditional

probability to calculate that the likelihood of those three cases is not the same.

As I tried to show in my previous comment:

Case 1. P(The coin toss was heads & the day is Monday) = .5

Case 2. P(The coin toss was tails & the day is Monday) = .25

Case 3. P(The coin toss was tails & the day is Tuesday) = .25

She should not "count the number of awakenings that happen in the heads timeline relative to the total number of awakenings" , that would work if the likelihood of each of the three categories of awakenings is the same, which it is not.

She can't know which of the three cases holds when she is awakened, but she can calculate the probability of each case. When she does that correctly her degree of certainty that the coin landed heads should be .5 .

I agree that if she gets to bet a dollar every time she awakes she should bet on tails. That's because she gets to bet twice when it's tails (Monday and Tuesday) but just once (Monday) when it's heads.

When she bets tails

50% of the time she loses $1 on Monday: when it lands heads

50% of the time she wins $1 on Monday and also wins $1 on Tuesday: when it lands tails

Hence her expected return betting tails is

.5*(-$1) + .5*($2) = $.5 when she bets tails.

When she bets heads

50% of the time she wins $1 on Monday: when it lands heads

50% of the time she loses $1 on Monday and also loses $1 on Tuesday: when it lands tails

Hence her expected return betting is

.5*($1) + .5*(-$2) = -$.5 when she bets tails.

So she should bet tails but

She still expects to see heads and tails 50-50.

Sorry, I found a mistake in my previous post.

At the end of the section which begins

"When she bets heads"

I mistakenly said:

.5*($1) + .5*(-$2) = -$.5 when she bets tails.

I should have written:

.5*($1) + .5*(-$2) = -$.5 when she bets heads.

@Patrick,

You've missed my point entirely.

For a moment, forget all the calculations and concentrate on just this one little point – how can you determine someone's "degree of certainty"? After all, it's just something that's inside your head.

According to the Bayesian "Dutch Book argument" I mentioned, "a degree of certainty" or "degree of belief" or "credence" is, simply, your

willingness to wager*. Specifically, if you have a "degree of certainty" of 1/n then you should be willing to accept a bet that offers you n or more dollars for every dollar you bet. This is simply common sense. (Of course, it is assumed that you want to win the bet.)So if, as you say, Sleeping Beauty should have a degree of certainty of heads of 1/2 , she should be willing to accept a bet that paid her a total sum of more than $2 on a $1 bet on heads. So she should be up for the following bet: She puts out $1, and I pay her $2.50 if the coin had turned up heads. But if she takes this bet, she will lose in the long run. Her outlay over two coin tosses would be $3, and she would win $2.50 just once.

She would break even on such a bet only if she were offered $3 on a $1 bet. Ergo, her degree of certainty of heads has to be

1/3.This definition of degree of certainty or credence is a perfectly good, usable, common sense definition from an individual point of view, and moreover, its value can be determined precisely in an objective way.

You will say, this definition of degree of certainty is not what you had in mind. Actually, believe it or not, you did! Your definition of degree of certainty was actually just the same, but it was from the experimenter's point of view.

BTW, your statement:

>She still expects to see heads and tails 50-50.

is wrong. Only the experimenter will expect to see heads and tails 50-50. She will "see" heads and tails in the ratio 1:2.

I'd like all confirmed halfers and thirders to read this column carefully, go through the variations and and try and feel the other point of view that does not come naturally to you. That way lies world peace 🙂Cheers!

Pradeep

*http://plato.stanford.edu/entries/epistemology-bayesian/

“Halfers take an experimenter’s view: For them, the question is about which of the two arms of the study Sleeping Beauty is in”

Well exactly. Despite the problem statement clearly instructing them to take SB's view *and which interpretation of probability she and they are to adopt* they don't and so they get the solution wrong. Clearly, plainly, obviously, simply wrong. There just is no ambiguity in this – https://en.wikipedia.org/wiki/Sleeping_Beauty_problem#The_problem – and nor is there any subtlety about what information can/should be used (as you point out: if one doesn't use all of it, the problem becomes trivial – a fact which is itself a profoundly unsubtle hint!). So I'm afraid I can't see any reason to respect the halfers' wrong answer (or the extensive literature of excuses contrived in its defence).

Very interesting article, I enjoyed the Necker Cube metaphor and the idea of an "halfer/thirder" psychological research project.

Still, the problem seems to me fruit of a misunderstanding. Sleeping beauty should answer: «I get awakened every Monday and 1/2 Tuesday. So my degree of certainty that the coin toss is head for every session is half-half, but I will get awakened twice in the same session if it's tail.»

The "bet arguments" are a sort of "cheating", because she would bet twice in the same session: if she knows it, she would refuse the bet, but the degree of certainty about the coins will remain the same. As L. Smolin put it: "Mathematics is a great tool, but the ultimate governing language of science is language.".

I'd like to thank Pradeep for featuring the jellybeans variants of SB in this article, and all commenters (not least of all P himself!) for lively debate 🙂

My motivation for concocting variant 1 was to understand how SB's credence should influence her decision-making. Picture her there, drowsily trying to figure out whether she's going to live to tell the tale of her choice(s) … "there's an x% chance of Heads, in which case I have a y% chance of survival …". But what "x" to use in those calculations? Well, it's 1/2, as Pradeep has shown. And that "x" is necessarily her "credence now for the proposition that our coin landed Heads" — verbatim from 1999 (http://www.maproom.co.uk/sb.html) — right? No, say Thirders, it isn't.

You might think that the same problem afflicts Halfers in variant 2. Not so. Halfers and Thirders alike agree that the probability of the *most recent* coin flip being Heads is about (Halfers) or exactly (Thirders) 1/3. In that "x% Heads, y% survival" calculation, we're substituting our credence for "x", just like Halfers did in variant 1.

I thought all this was a clear-cut demonstration of Thirder irrationality, basing their decisions on credence only when it suited them, until I came across the Absent-minded Driver problem, which predates SB.

In short, AMD goes as follows (again, see http://www.maproom.co.uk/sb.html). Our driver is on a motorway with two potential exits. Taking exit 'A', he'll end up in bad part of town (this has zero utility); taking exit 'B', he'll get home (utility 4); and missing both exits, he'll stay the night at a motel (utility 1). The exits are indistinguishable and, absentminded as he is, he won't be able to remember if he's already gone past an exit. The correct strategy in this case is a "mixed" one: at any exit, you make a snap decision to turn off with probability 1/3. However, once on the road, you will have a non-trivial credence for being at exit 'B', on which basis the optimal exit probability would appear (wrongly!) to be greater than 1/3.

The details of AMD are rather fiddly to work through, but the conclusion is nonetheless straightforward and potentially undermines for SB variant 1: choices based on momentary credence (like when SB wakes) may be suboptimal! Thirders can point to this as a potential defence for not plugging x=1/3 into that chances-of-survival calculation.

Of course there is a Halfer counter-argument to that defence, and probably a Thirder counter-counter-argument, and so on. It all goes to show just how nebulous this concept of "credence" is.

Hi Pradeep,

Thanks for elaborating further. Statements and questions written informally in natural human languages are notoriously prone to ambiguity and multiple (mis-)interpretations. My research specialty for many years was the formal semantics of computer programming languages. Unsurprisingly that field goes to excruciating lengths to adhere to conventions which minimize those types of problems – many of the techniques were borrowed from mathematical logicians like Alonzo Church and Haskel Curry. Even in that austere and rigorous environment mistakes and ambiguities sometimes crop up, especially in tricky areas like the conventions for formal variables.

Your exposition of the Sleeping Beauty Dilemma and the thoughtful comments of the other readers have been most stimulating and enjoyable, thanks!

Also you pointed out:

BTW, your statement:

>She still expects to see heads and tails 50-50.

is wrong. Only the experimenter will expect to see heads and tails 50-50. She will "see" heads and tails in the ratio 1:2.

When I said "She still expects to see heads and tails 50-50" I was thinking of the actual physical coin toss at the beginning at the of the experiment and that – if the experiment were repeated multiple times – at the end of the experiment on Wednesdays she will expect to find out that the physically tossed coin landed on heads 50% of the time.

However, upon rereading the original statement of the problem, it never explicitly says that she will find out the outcome of the physical coin toss on Wednesday. I just assumed that, which is is probably not be justified.

That obviously wasn't the thrust of your remark, but it ended up helping me anyway, thanks!

@Patrick

Thank you for clarifying what you meant when you said "She still expects to see heads and tails 50-50."

Your reasoning is absolutely right. When Sleeping Beauty wakes up on Wednesday, blissfully blank about what happened in the past two days, she is back in the experimenter’s world, so she should agree with the experimenter’s degree of certainty about heads: one-half.

We can picture them on Wednesday morning at the exit interview: SB and the experimenter, back from his conference on Bayesian epistemology and also unaware of the result of the coin toss. On the table before them is a sealed envelope put there by the assistant who carried out the experiment, revealing how the coin actually came up. Both SB and the experimenter agree that they would accept a bet that offered more than $2 that the note in the envelope says “Heads.” This reflects the fact that their credence for heads is now one-half. Then they open the envelope and find out what happened. As in quantum mechanics, their credence settles down on a value of either exactly 0 or exactly 1.

What does this tell us? Degrees of certainty (credences) change depending on the state of your knowledge at a given moment. On Sunday, SB’s credence for heads was ½. On Monday/Tuesday, cocooned in the lab, her credence for heads was 1/3. On Wednesday on awakening, it was back at ½. After reading the result, it settled on 0 or 1. This is at it should be, and our definition of a credence = 1/n as the willingness to make a winning unit bet offering more than n dollars, makes the concept objective and precise.

@Francesco

I understand your vague sense that there is some “cheating” here. The above scenario tells us where that comes from. At the back of SB’s amnesiac-addled mind, she knows that she will come back to the real world and her credence for heads will be back at one-half, and she will make a different bet. I think this is the core of the halfer’s intuition – that somewhere, back in the real world, the credence is one-half. But at the moment of awakening, when the question is put to SB, using all the information that she can bring to bear about the time and the situation, her credence for heads, using the above operational definition, is indubitably 1/3.

@eJ

The operational Dutch Book definition of credence I’ve discussed with Patrick above is not nebulous at all: it is precise and objective.

Here’s how it works for Variation 1. There are some problems for which you have to go back to the credence that you had at a certain time in the past, and for Variation 1 you have to go back to the credence you had on Sunday.

Let me give an example to clarify. Suppose, when you were ten years old, you wrote a note addressed to yourself 10 years in the future. You hid it in one of two secret places in your house and resolved to find it when you turned twenty. Today is your twentieth birthday. You can’t quite remember where you hid it. Today you can think of three places where you would hide such a note – so your credence for each is 1/3. But then you remember that one of those places is in an addition to your house that was put in only when you were 15. Now you need to do what we can call a “credence reversion” or a “credence pop.” You have to pop your past credence off the stack, and reassume it to get the right answer.

That’s what a thirder has to do for Variation 1. Halfers, of course, stick to their Sunday experimenter view of credence forever. For Variation 2, thirders just use their present credence. Halfers, on the other hand, calculate the present credence from their default value using probabilistic principles. They are in effect doing the opposite of the above –a “credence push” instead of a “credence pop”. They use their temporary new credence to solve Variation 2, and then pop it out again. Smart halfers and thirders can both recognize what procedure they need to apply in a particular specific problem. Both work – it’s down to your personal preference.

Does the operational view of credence survive the Absent-minded Driver problem? Well we’ll just have to do a “credence suspension” on that and perhaps wait for a future Quanta Insights column, won’t we?

@phayes

Strong words, bro! I know where you’re coming from – I’m from there myself. But I think being a dove and listening is more productive. You tend to build a more sophisticated, nuanced and precise model that way. Would that would happen in politics!

It is a fact that the definition of “degree of certainty” is left ambiguous in the problem, so halfers are free to do what they do. But I agree with you that halfers are choosing to answer the uninteresting and trivial question in the SB problem. As can be seen in the discussion above, they do have a strong intuition that they base it on, about what SB’s credence should be in the real world in which the experiment is embedded. Thirders are like perspective artists – they look at the world from a given point in time and space. Halfers prefer to eternally enjoy the out-of-body view from above.

There is also the fascinating question about what information SB is getting about the coin toss. Is she getting information? As I described above she is getting diffuse situational information from three sources. None of this information is about what actually happened during the coin toss. In this scenario, SB is compelled to forget or suspend her knowledge of the real world. She has to answer based on the situational knowledge from within the experimental scenario. Halfers are stuck on the fact that this is not real world knowledge. They cannot, or will not, change their credence to fit the situation.

Now we wouldn’t be able to discuss all these fascinating questions without our halfer frenemies, would we?

Cheers!

Pradeep, by offering that credence was a "nebulous" concept, I was simply trying to give flight to that dove you mentioned. We can all construct precise definitions of credence, but the correctness of any candidate definition is hard to pin down.

For example, we both have betting tests (beans and "Dutch Books") for SB's credence, but no-one's offering a compelling, Necker-cube-flipping, reason to accept one over the other.

@eJ,

Nah, this Necker cube doesn't flip – I think the tendency to be one or the other may be built too deeply into our personal mathematical intuitions. Didn't you see the tabloid headline:

"Halfer changes to Thirder – World To End on Wednesday"

@eJ,

There's a slight difference between the Dutch Book definition and the beans test. The Dutch Book definition applies generally in any situation you can think of – it is just a method for an external observer to quantify someone's credence objectively.

The beans test, on the other hand, is specific to the SB problem. Do you (or any of the halfers here) want to give a simple general definition of credence that supports the halfer position? I know you treat time differently from the way I explained it in my earlier post.

Thanks!

“It is a fact that the definition of “degree of certainty” is left ambiguous in the problem”

It's a fact that it isn't – I refer you to the Wikipedia page again – and the true nature of the halfer excuse is also made clear on that page (albeit in an irrelevant and wrong-headed 'operationalization' context):

“However, being fully aware about the experimental protocol and its implications, Beauty may reason that she is not requested to estimate a statistics of the circumstances of her awakenings, but a statistics of coin tosses that precede all awakenings. She would therefore answer P(Heads) = 1/2.”

“We're not the ones unable or unwilling to adopt the probability interpretation which we've been instructed to adopt – Beauty is!”

@phayes

The SB problem simply states "During the interview Beauty is asked: "What is your belief (subjective probability, credence) for the proposition that the coin landed heads?"

It does not specify how belief or credence is quantified, nor does it specify whether the proposition that the coin landed heads is to be evaluated within the sealed environment inside the experiment (as thirders do) or in the outside world (as halfers do).

So again, halfers have every right to interpret the problem the way they do.

Your wikipedia quote merely describes one way in which Beauty may reason to come up with the halfer answer in the author's analysis. It is something that is

notspecified within the problem. Also, note that the method described does use the "probability interpretation" – it just uses "the statistics of coin tosses that precede all awakenings" instead of the statistics gathered by Beauty herself.So again, the procedure is perfectly legal in the context of the question.

I think you should stick to the claim that halfers are answering a trivial question whose answer (fair coin) was already provided in the problem statement. That much is true.

@eJ

On further reflection, I don't think that halfers and thirders disagree about the quantification of credence – they both implicitly or explicitly use the procedure I described, which is properly called an

operational subjective probability. (The Dutch Book is a more complicated version of this and that term should not be used in this context. See this link)Where halfers and thirders have a fundamental difference, as I stated above, is whether

SB should evaluate her credence for the proposition within the sealed environment of the experiment where she finds herself (thirders) or whether she should evaluate it by projecting herself in an out-of-body way into the world outside the experiment (halfers).“It does not specify how belief or credence is quantified,”

Well it doesn't specify whether to use the de Finetti or the Cox-Jaynes way of quantifying belief [as probability] but what difference does that make?

“nor does it specify whether the proposition that the coin landed heads is to be evaluated within the sealed environment inside the experiment (as thirders do) or in the outside world (as halfers do).”

It certainly does. “During the interview Beauty is asked: "What is your belief (subjective probability, credence) for the proposition that the coin landed heads?"”. Beauty is asked. Not you or I, not the experimenter – Beauty. And she is asked what is *her* belief that the coin landed heads. Not ours, not the experimenter's, not her belief about someone else's belief or what her own belief might be in different circumstances. She is asked for her belief – her ('subjective') probability assignment – there and then for the proposition that the coin landed heads.

Pradeep, you asked for a definition of Halfer credence. See https://www.quantamagazine.org/20160114-sleeping-beautys-necker-cube-dilemma/#comment-364394 (yuck, how do I get nicer-looking hyperlinks?). Thus my credence for X is the expected value, over each random outcome of an experiment, of my credence for X given that outcome.

This is *not* the definition used by Lewis (the original Halfer, see his "reply to Elga" paper), but instead reflects the "Double Halfer" position of Bostrum, Cozic and others. (We've not been clear about the halfer/double-halfer distinction up till now.) Specifically:

* Cr(Heads) = 1/3 (thirder), 1/2 (halfer), 1/2 (double halfer)

* Cr(Monday) = 2/3 (thirder), 3/4 (halfer), 2/3 (double halfer)

* Cr(Heads|Mon) = 1/2 (thirder), 2/3 (halfer), 1/2 (double halfer)

I dislike your "out-of-body" snipe at the [double] Halfer position. We disagree on what constitutes an experiment, i.e. the procedure that gives rise to a https://en.wikipedia.org/wiki/Probability_space . Thirder calculations imply a probability space of three mutually-exclusive equiprobable outcomes, { Heads, Tails/Mon, Tails/Tue }. What experiment produces a single outcome as described by that probability space? Can you perform a single run of that experiment in the lab?

Thank you for your kind reply and for the stimulating debate 🙂

You write (and I agree):

"SB should evaluate her credence for the proposition within the sealed environment of the experiment where she finds herself (thirders) or whether she should evaluate it by projecting herself in an out-of-body way into the world outside the experiment (halfers)."

If we define her "credence" as "the probability to guess the right answer in the real world", the interpretation of halfers sounds again the best. In a metaphysical way, both thirders and halfers can be right, since our difficulty to define reality (I will not open this interesting but huge parenthesis) but the world of the riddle is simpler than ours, it's "given" and we accept it with the game itself. In this scenario, the real world is the one of the halfers and the thirders world is a subset of it.

In the end this sounds not as a mathematical enigma but aa an ambiguity of the natural language term "credence". Even in Necker cube the paradox works due the ambiguity of the depiction rules of a cube – it could not even be perceived as a cube if the beholder is not used to perspective

@phayes

Whenever I reply to you, I’m tempted to paraphrase a famous quote by Evelyn Beatrice Hall, “I may not agree with what the halfers say, but I’ll defend their right to say it.” Maybe not to the death, though ☺.

Your point about it being SB’s belief and no one else’s is absolutely true, and it may seem pretty clear-cut what it should be at first sight. But imagine this conversation between the sophisticated and intelligent princess SB, and the experimenter’s assistant who interviews her on her awakenings.

EA: What is your belief for the proposition that the coin landed heads?

SB: That question is ambiguous. Do you mean my belief about the result of the Sunday coin toss that you are carrying in your head, or do you mean my belief in the result of the Sunday coin toss that I will find out on Wednesday?

EA: The result that I know took place.

SB: My belief that was heads is one-third.

EA: And what about the result of the Sunday coin toss that will be revealed to you on Wednesday?

SB: One-half.

EA: Perfect! Right on both counts!

If the SB in our little story had cleared the ambiguity right there, we wouldn’t be having so much fun!

@Francesco

Thanks to you too for your great points!

Here’s what I’d like to add to your statement: “If we define her "credence" as "the probability to guess the right answer in the real world", the interpretation of halfers sounds again the best.”

…And if we define her "credence" as "the probability to guess the right answer from her own point of view", the interpretation of thirders sounds the best.

It is SB’s world that is a subset of the game’s “real world” and that too, only when she is a subject, on Monday and Tuesday. She starts in the game’s real world on Sunday, and reemerges into it on Wednesday when the experiment ends. Thirders merely follow it through her eyes because the question has to be answered from SB’s point of view.

I completely agree that both can be right, and it is the ambiguity in the question that makes both interpretations possible. That being said, I agree with phayes that the halfer view makes the problem trivial.

@eJ

Okay, the credences you give can be easily understood by understanding whose statistics they reflect.

Sleeping Beauty collects statistics: Thirder. SB encounters Heads 1/3 times, Monday 2/3 times and on Mondays she encounters Heads 1/2 times.

Experimenter collects statistics: Halfer. The experimenter only attends one interview session per experiment. If the coin toss landed heads, she has to be at the Monday session. If it landed tails, she chooses Monday or Tuesday at random. She encounters heads ½ times, she’s there on Monday 3/4 times, and on Monday she encounters Heads 2/3 times.

Assistant collects statistics: Double Halfer. The assistant attends all the interviews, but tallies the tails only the first time, because it is the same experiment. He encounters Heads ½ times, but the other two numbers are the same as SB.

My “out-of-body” statement was not intended as a snipe. I was imagining her leaving the lab in her mind, and looking on from outside, instead of responding based on her current situation – a second order act on her part.

The experiment that produces the outcomes is Snow White’s experiment – remember, we are looking through her eyes. A single run of that experiment is one awakening. The equiprobable outcomes (independent and indistinguishable to her, because of her amnesia) are: {Heads-Mon, Tails-Mon and Tails-Tue}. She does not sample them randomly: TM is always followed by TT and TT cannot occur by itself, but she doesn’t know that. It does not make any significant difference* to the statistics she collects, which are the same as she would get by random sampling.

*[Actually the non-random sampling does make a very slight difference, because for the first sample, the probabilities are 50% HM and 50% TM, but very soon the H:Total ratio converges to 1/3. As we saw, the error is of the order 10^-31 for one hundred awakenings.]

I find it a little ironic that halfers don't put themselves in SB’s place and give any importance to the statistics she collects herself, on which she would be expected to base her first-order reply. And yet, they would like her to answer the credence question as though she were in the experimenter’s place!

“But imagine this conversation between the sophisticated and intelligent princess SB, and the experimenter’s assistant who interviews her on her awakenings.”

Okay but then I imagine the Chief Experimenter's reaction on overhearing that exchange…

CE [to SB]: Oh dear! This is most embarrassing. As well as seeing ambiguity in our carefully designed experimental question you've revealed that you believe that it is less likely that “the result that EA knows took place” will be heads than will “the result of the Sunday coin toss that'll be revealed to you on Wednesday”. But they are the same result and one of the inclusion criteria for participants in this experiment is that they be “ideally rational epistemic agents”. So I apologise – the fault is ours – but I'm afraid you'll have to be withdrawn.

CE [to EA]: And you are fired.

😉

No, that experiment "in the lab of her mind" is no good. You can't run any number of independent instances of such a thing. As you said yourself: she does not sample (from the state space) randomly; and she always gets two outcomes, not one, in strict sequence on Tails. That does not comply with the definition of an experiment in the sense of probability theory.

In this puzzle, Beauty is asked about a genuine honest-to-goodness probability event, namely the outcome of the coin flip. Yet again: "credence now for the proposition that our coin landed Heads". It doesn't "land Heads" in her mind. You know, in Beauty's position, that the probability of landing Heads is 1/2, and the circumstances of your awakening do nothing to change that, so there is certainly some question along the lines of "what's the chance of Heads" to which you'd answer 1/2 (if for no other reason than to justify your choice of bean bag). Yet you choose to parse the apparently-synonymous "credence for Heads" differently, prompting the answer 1/3. It's all very strange.

We have become entrenched, and it is late. I'm taking at least 24 hours off this thread as of now.

@phayes

Oops, I'm embarrassed! You're right. That dialogue as I wrote it was a disaster. I was trying to express the intuition that if SB were asked to guess on Wednesday what her belief for heads was, she would have to say one-half.

But that would mean that SB is doing second order out-of-body projection of her mind to a different time point where she has forgotten what she knows, which is kind of iffy…

Okay, I'm done defending halfers! For now, at least. Let them do so themselves. Based on her statistics and her situation, SB's only honest-to-goodness answer from her own point of view has to be 1/3. It is possible for her to do some mental gymnastics and imagine how the coin came up in the world of the experimenter, but that would entail her taking on a mental state where she has to forget what she now knows.

I'm with you, bro! I'm a thirder and taking on the mantle of a halfer is hard. I guess my Necker cube that I was trying hard to flip, is back in its original position.

@eJ,

I agree, we should take some time off. Maybe someone else will chime in.

The one fleeting thought I have in response to your last post is that just because a population of events does not conform to a pure model with random sampling doesn't mean you can't do probabilities with it. You can always use frequencies to get the right answers. SB's statistics are clear-cut. She doesn't actually have to collect them, she can compute them theoretically based on her knowledge. Based on that she can make probabilistically accurate predictions about her chance to encounter heads. And it is 1/3.

@eJ and phayes,

I think I may have pinpointed at least one source of linguistic ambiguity in the SB problem. It may go a long way towards explaining how halfers and thirders (or least some of them) may be talking past each other.

Imagine that when the coin was tossed on Sunday, it was made to land on a platform to which it was stuck with superglue, so that its position was frozen for eternity.

Now the original question “what is your credence that the coin landed heads” can be interpreted in two different ways:

1. What was your credence that the coin landed heads when it was originally tossed? (This question may even be understood the same way when phrased in the present tense: what is your credence…)

2. What is your credence that the preserved coin is showing heads now?

SB’s answer to the first question, of course, is ½ and remains 1/2 throughout. Her answer to second question is 1/3 at the time of her Monday/Tuesday awakenings, and changes to ½ on Wednesday when the experiment ends, before she is told the answer.

When I look back at eJ’s posts, it seems to me that he always has SB answering question 1. I, on the other hand, have always had her answering question 2 except in Variation 1, where it is obvious that question 1 is the relevant one.

What do you guys think? Does this resolve our differences? Can we part amicably now in complete agreement?

Somehow I don’t think so 🙂 but maybe it helps.

Well I (still) think there is no way to justify the halfer solution. The statement of the SB problem just doesn't leave enough room for it. I almost wish it did. 😉

@phayes

Now you are just being uncharitable ☹. As we saw, both halfers and thirders reach the exact same correct solutions when faced with SB problem variations that are completely specified. This problem would not have remained open for over 15 years unless there is some ambiguity in its statement that smart people can disagree about.

After two days of mulling over it, I am convinced that my last post uncovers an important reason for the ambiguity.

To repeat, the original question “what is your credence that the coin landed heads” can be interpreted in two different ways:

1. What was your credence that the coin landed heads when it was originally tossed?

2. What is your credence that the preserved coin is showing heads now?

You and I intuitively choose the second alternative, which entails counting the frequency with which SB encounters heads relative to all her awakenings.

Halfers, including eJ, Pace and Peter Winkler, all of whom appear extremely smart, intuitively focus on the first alternative, which can also be the way SB may choose to understand it. In that case her statistics become irrelevant.

So though the Necker cube has not flipped in my mind, I can appreciate that it may be seen from two perspectives, and is not one with the front face painted blue.

@eJ and all other halfers,

This problem fascinates me, and I undertook a

self-imposedmission to present it in a completely neutral way. This is different from most of the treatments available on the web, the majority of which are unabashedly partisan. I think I did succeed in being neutral in the text of the blogs.However, the thirder view comes naturally to me. In spite of trying to bend over backwards to be neutral, I could not hide my intuitive enthusiasm for the thirder point of view in my comments as an “ordinary citizen.” Unless I clearly labeled the comment “Author’s update,” I was not attempting as hard to be neutral here. I debated using a pseudonym for my comments, but in the end I submitted to the famous “Hillary Clinton temptation” – the convenience of having just one Blackberry! In the end, I think it is more honest this way.

Even though I am intuitively a thirder, I am intellectually an “ambiguist.” As I stated above, this problem would not have remained open for over 15 years unless there is some ambiguity in its statement that smart people can disagree about. Therefore, I encouraged people from both camps to contribute their views, and still do. It’s only by trying to understand the views of people in the other camp that we can hope to perceive both the views of the Necker cube. So if I bubbled over to my intuitive side in the heat of the debate and put off any halfers, I apologize. Please feel free to contribute – going after my views is fair game too 🙂 .

Cheers!

"Now you are just being uncharitable"

Charity is irrelevant. As is the (duration of) disagreement between smart people. A clever / elaborate post hoc rationalisation is still a post hoc rationalisation. You say the question of the SB problem statement can be interpreted as "What was your credence that the coin landed heads when it was originally tossed?". I say "is" cannot be interpreted as "was".

One more thing… One doesn't have to count frequencies, or justify probability assignments the de Finetti way, of course. I prefer the Cox-Jaynes way¹ (and I suspect Patrick O'Keefe would too).

¹ E.g.

Pr(H|S) = Pr(H|Mon,S)Pr(Mon|S) + Pr(H|Tue,S)Pr(Tue|S)

= (1/2)Pr(Mon|S) + 0 [by the fair coin assumption]

Pr(Mon|S) = Pr(Mon|H,S)Pr(H|S) + Pr(Mon|T,S)Pr(T|S)

= Pr(H|S) + (1/2)(1 − Pr(H|S)) [by the principle of indifference]

= (1/2)(Pr(H|S) + 1)

Pr(H|S) = 1/3

Pradeep, in the hope of shedding a little more light on the nature of the Halfer/Thirder disagreement, I'd be grateful for the Thirder view of the following argument.

Beauty's credence for Heads is the same on Monday as it is on Tuesday (she's not told which day it is), so let's set ourselves the task of measure her credence for Heads on (not /given/) Monday. On Monday, we'll offer a wager: she wins $3 if Heads, she loses $2 if Tails. So that this gives her no information about the day of the week, we'll offer the same wager on Tuesday but (by agreement in advance of the experiment) there will be no payment to either side for that instance of the bet. Of course Beauty should take the bet, How does Thirder credence motivate that choice?

Fantastic column and excellent arguments/counter-arguments.

Though I understand the concept of probability and its application I always get woozy when it gets to specific circumstances.

Though I read all the above expert arguments I still feel she would be saying 1/2 and the reason that I came up for this is as follows-

Since SB is oblivious to her awakenings but know the rules of the game, the state space available for her when she awakens is I feel as follows: {MH, MT, Tue}

The reason I don't give a coin flip for Tuesday is because if she wakes up on Tue then it's foregone conclusion that it came up tails.

My conundrum is as follows, she has more probability of waking up on Monday so then wouldn't her choice of heads be just 1/2? I am not a math person but an engineer so I could have overlooked many things here or my assumptions itself plain wrong.

Her correct answer is "zero."

Wow, this was such an awesome thought experiment. Thank you, Pradeep, eJ and the other fantastic commenters. This one has kept me thinking continuously over the past two weeks. I know I’m a little late to the game, but I’d like to throw in my lot with the halfers.

Hopefully nobody minds too much that I’m going to be changing the thought experiment slightly, just to stretch things out a little more—it helps me visualize what’s going on inside the experiment a little better. This variation on a theme is a kind of combination of the two variations listed in the puzzle above. With that in mind, I propose stretching the experiment out from three days to one week. It doesn’t make much of a difference—three days will still work, but, again, seven days helps with the visualization. A hundred days, as in variation 2 above, works just as well, but let’s just start with a week (Monday to Sunday). Another thing before I move on to my scenario, I’m going to be stealing some of eJ’s jellybeans and poison pills.

Okay, as usual, Sleeping Beauty is going to be put to sleep on Sunday and a fair coin will be flipped. If the coin comes up Heads, SB will be woken on Monday and will choose to draw once from a bag labeled Heads or Tails, each bag contains 6 jellybeans (J) to start. This is modified from eJ’s scenario in that there is just one Heads bag and one Tails bag—the contents of the two bags no longer change depending on the coin flip. One other big change: Each time a J is drawn from a bag, a poison pill (K) replaces it. On Monday we’ll start both bags with 6 Js each.

So, back to SB, if the coin comes up Heads, on Monday she’ll choose either the Heads or Tails bag, reach in and pull out a J. She’ll put the J aside, take her “forget-me-now” and then be awakened at the experiment’s end on Sunday where she’ll eat the jellybean. Conversely, if the coin flip comes up Tails, SB will be awakened on each day of the week (Mon-Sat), each day choose a bag, put the J or K aside, take her “forget-me-now”, repeat until experiment’s end on Sunday where she’ll have to eat the 6 Js/Ks.

Clearly it sucks to get a Tails flip in this scenario—Beauty is going to have many more chances at drawing a poison pill. Is there a strategy that SB can use to minimize her chances of getting a K? Yes! The only strategy in this case is to be an uncompromising halfer. Here’s why: If SB decides that she’s going to go into this experiment with any other kind of belief or strategy (thirder, or in this case seventher), or if she goes into the experiment with some kind of unknown bias for Heads or Tails bags, she’s going to get herself killed at least 83% of the time—5 out of 6 possibilities on the final draw on Saturday will be Ks.

The reason for this is that each time Beauty wakes she is going to remember whatever her initial strategy/bias was going into the experiment and she’s going to risk continuing to make the same decision based on her credence each and every day. Remember the amnesia is going to make her forget each awakening, so she is never going to receive any information that will make her want to change her actions. She will most likely continue to draw from the same bag each day—every day will seem like Monday to her within the experiment. The only way to avoid sampling from the same bag over and over again throughout the week, thus increasing its K count, is to fluctuate between the Heads and Tails bag as much as possible. The best way for her to do that would be flip her own coin (which we will kindly provide) each time she wakes. Sleeping Beauty has just become a halfer. She may still feel like it’s more likely she’s waking on the Tales branch, but that’s really just an illusion. Her life depends on her being a halfer and flipping her coin each time she wakes.

It doesn’t matter how many times she wakes up throughout the experiment, it doesn’t change the 50/50 probability of the initial coin flip. We can all agree that when the experiment ends on Sunday (or whatever day we choose) Sleeping Beauty is going to be a halfer at that point: 50% chance of seeing 1 J or 50% chance of seeing 6 J/Ks (Heads or Tails).

I apologize if this comes across as a tautology, but it seems necessary to avoid the ambiguities of this problem. A lot of those ambiguities can be resolved with specifying whatever the question is, but I think some might just be the result of information leaking into the thought experiment. Sometimes we might lose focus and give SB more information than she is entitled to.

Paperclip Minimizer, that's an interesting scenario. Unfortunately, I don't think it reveals anything about Beauty's credence for the coin flip, or indeed the inherent bias (if there is any) of the coin. Irrespective of the objective probability of ending up in one-wakeup or six-wakeups world, Beauty should pick a bag at random each time.

@eJ,

Ah, but by making the Tuesday bet a dummy bet, you are, from the thirders point of view, changing the experiment.

As I’ve explained before, here’s how the thirder reasoning process works.

Thirders, like SB and the halfers, were also halfers on Sunday. This is obvious and trivial – it is a fair coin after all.

On Monday and Tuesday, thirders (and we argue, SB) update their credence for heads to 1/3. For this to happen, the following conditions must be met: 1) The 3 possible awakenings must be indistinguishable to SB; 2) The three awakenings must all be given equal weight – for SB, this follows from condition 1; and 3) Only one of the three awakenings must be associated with heads.

These three conditions are met in the original experiment.

This gives rise to the sample space {Awake-H, Awake-T, Awake-T}. From the experimenter’s point of view, this is actually {HM, TM, TT}, but from SB’s point M and T are indistinguishable. This makes her credence for H equal 1/3.

By making the Tuesday bet a dummy one, you are violating condition 2. You are changing SB’s known sample space to {Awake-T-1, Awake-H-1, Awake-T-0}, which normalizes to {Awake-T, Awake-H}. So in this case, when SB updates her credence, she remains a halfer. She is never a thirder in this new experiment.

(Incidentally, this is also what happens in Variation 1: the Tuesday beanbag is essentially irrelevant to the life or death question).

(Interesting parallel with Quantum Mechanics here: in the double slit experiment, if you remove the indistinguishability of the two slits, the experimental outcome changes.)

Don’t you see that the thirders do the same calculation that the halfers do? It’s just that they do the calculation, hold it in readiness and call the result their credence. They define credence to mean their readiness to accept what I would call an “existential belief” bet. If you are willing to accept a bet on the existence of the outcome at odds of n to 1 or better, your credence is by definition, 1/(n+1).

This is a definition of credence that is absolutely general and can be applied to any situation whatsoever. If someone asks me what is my credence in the possibility of rain tomorrow, I ask myself what odds I would accept that it will rain tomorrow. If I think there’s a 20% chance of rain tomorrow, I am fine with odds of 4:1 or better and my credence for rain, by definition, is 1/5. It’s that simple.

In the case of the rain, there was no previous credence that I had to update. In the SB problem there was a previous credence of ½ that needed updating. The difference between thirders and halfers is when they do the update. Thirders do it continuously based on the situation, kind of holding it in readiness for an existential belief bet. Halfers only do it when challenged with an actual bet. For halfers, their previous credence is their current credence until challenged by a bet. But in many situations, there is no previous credence, so the halfer and the thirder definition of credence would be exactly the same.

This is why I commented that halfers may be temperamentally cautious, while thirders may be temperamentally pragmatic.

@Hemanth

As I described above, SB does not know when she is awakened whether it is Monday or Tuesday. So your state space of {MH, MT, Tue} cannot be correct from her point of view.

Her state space is {Awake-H, Awake-T, Awake-T} or even more strictly {Awake-1, Awake-2, Awake-3} where only one out of the 1, 2 and 3 is H. Only later will she find out which one it actually is. So the subjective probability of heads from her point of view, is 1/3.

Pradeep, for the time being I am interested only in the specifics of the Phantom Tuesday variant (to give a name to the no-payment-on-Tuesday thing from February 4, 2016 at 3:33 am).

You have said that in Ordinary SB, her credence for Heads is 1/3, but in Phantom Tuesday SB, her credence for Heads is 1/2. Without a digression into the wider world of jellybeans, quantum mechanics and the weather — and let's say without any mathematics at all — can you give any heuristic that justifies how the structure of the wager (and that alone, with all per-wakeup observables being as normal) has any bearing on her credence for Heads?

I don't know if this has been answered already (my previous post is still the last one showing), but I'd like to close off the Phantom Tuesday thing. Thirders would say E(winnings from this wake-up) = 3.Cr(Heads,Mon) – 2.Cr(Tails,Mon) + 0.Cr(Tails,Tue) = 1/3, so would quite correctly take the bet. Double Halfers would say E(winnings from the whole experiment) = 3.Pr(Heads) – 2.Pr(Tails) = 1/2, also taking the bet. More generally in SB variants with independent cumulative bets, Thirder's this-wake-up expectation is just the Double Halfers' whole-experiment expectation divided by the average number of wake-ups: in ordinary SB: 1/3 = 1/2 / (3/2).

So we should stop challenging each other with scenarios involving independent cumulative bets. We'll always end up claiming Beauty makes the same choices.

We also all need to recognise that credence is an incomplete description of Beauty's mental state, insufficient to drive decision-making by itself. That is, if all you have are your credences, then there are scenarios in which you'll make the wrong choice. The Beans Experiment is a scenario that cannot be solved with (1/3,1/3,1/3) credences — Thirders must fall back to their prior belief state. Equally, Double Halfers have Cr(Monday) = 3/4, but we cannot use that credence to respond properly to a bet concerning the day of the week: we need to fall back to what we know about the experiment as a whole. Credence for X is not a basis, for either camp, to bet on X.

So if you can't always use your credence for X to place a bet on X, why worry about whether its value is 1/2, 1/3, or anything else? Let's be honest: neither camp can trick the other into making wrong decisions: our respective mental states are each quite capable of picking the right bean bag and making the right bet. Really all we're looking at with "credence" is a lossy compression of a complex web of probability estimates into just a single real number; it's how you choose to do that lossy compression that determines how you define "credence". It's not at all clear to me that there is, or needs to be, a right way to do that.

With that, I'm done. It's been an interesting, if often frustrating, discussion, and I will read with interest (but not respond to) any further comments. Good luck making sense of it all.

I'm disappointed to see Quanta publishing this "puzzle".

I know I'm a bit late but I had some thoughts and would like to weigh in.

This may have been stated before but this experiment is ambiguous and therefore irrelevant. Here's why:

The question that we all are trying to answer and the one proposed by the dilemma is "What should her answer be?" This is ambiguous because you can take this to mean two different things depending on your reading of "should." On one hand, you could read it as "What should her answer be if she is to have the best chance of guessing correctly?" Clearly here the answer is 1/2 because of the fair coin and the (already expanded upon) laws of probability. On the other hand, the question could also be read as "What should you expect her answer to be?" In this case you would answer 1/3 because that is what makes the most sense as a guess from a participant in the experiment's point of view.

The ultimate disagreement among the two sides is their interpretation of the word "should" in the question. Your own conjecture as to the meaning fundamentally shapes the way that you approach the problem and so it would seem that the two sides are answering two different questions in the end. That is why all of the arguing and trying to convince the other side is useless because in the end, the other side has a completely different view of the question proposed to them and the other answer does not make sense no matter how much explaining and calculation you do. I think that was the point of this solution article but it got ignored or misinterpreted by the impassioned commenters along the way.

This is the most important passage of the whole article:

"One reason for the differences between the two camps is that halfers and thirders interpret the question in two different, though equally valid, ways, and translate it into slightly different mathematical problems. Credence is an intermediate mental construct that is defined differently by the two camps. As Quanta reader Josh put it, “Like most verbal paradoxes, [this dilemma] relies on underspecifying the precise question!” When the question is put in precise mathematical terms, there is no paradox, and both halfers and thirders get the same answer — just as the ambiguity of the Necker cube is resolved if the front face is specified, as in the accompanying illustration. There can be clear-cut answers, then, as we shall see in our well-specified variations — there’s no need to worry about the foundations of probability. It’s just a matter of being specific about what is asked, as we saw in an earlier Insights column."