Is the answer to the Sleeping Beauty Problem 1/3?

The Sleeping Beauty problem is a puzzle in decision theory in which whenever an ideally rational epistemic agent is awoken from sleep, they have no memory of whether they have been awoken before. Upon being told that they have been woken once or twice according to the toss of a coin, once if heads and twice if tails, they are asked their degree of belief for the coin having come up heads.

Resolves based on the consensus position of academic philosophers once a supermajority consensus is established. Close date extends until a consensus is reached.


Small print

I will use my best judgement to determine consensus. Therefore I will not bet in this market. I will be looking at published papers, encyclopedias, textbooks, etc, to judge consensus. Consensus does not require unanimity.

If the consensus answer is different for some combination of "credence", "degree of belief", "probability", I will use the answer for "degree of belief", as quoted above.

Similarly if the answer is different for an ideal instrumental agent vs an ideal epistemic agent, I will use the answer for an ideal epistemic agent, as quoted above.

If the answer depends on other factors, such as priors or axioms or definitions, so that it could be 1/3 or it could be something else, I reserve the right to resolve to, eg, 50%, or n/a. I hope to say more after reviewing papers in the comments.

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Is there any example in recent history of a highly discussed philosophical question in which philosophers were divided for years into competing "isms" until a consensus was established which resulted in one of the positions being accepted as correct?

More generally, has philosophy ever provided clarity on any subject?

These are not rhetorical questions.

@MartinRandall This a much more substantive answer than I expected, and I am glad that I asked such a stupid question.
However, it's not clear to me that these examples really satisfy the conditions I asked for - a recent, entrenched dispute in philosophy that no longer exists, having been decisively settled.
To pick on just one example, number 7, this author claims that Hempel's raven paradox is settled, in this essay. But the Wikipedia article discusses numerous positions, and it appears there is some ongoing disagreement here after all.

@HarrisonNathan sure, this is just one philosopher's answer and it was more a reply to your second question than your first.

I am expecting available intelligence to increase over the coming years or decades so there may be faster progress in philosophy than has been the case historically.

@MartinRandall I probably shouldn't have phrased the second question the way I did: I meant for the "clarity" to be objective. Many people feel that the answer to the SBP is "clearly" 1/3 and others feel that it's "clearly" 1/2, but this is not what I mean. I would characterize the kind of things this author writes as the latter kind of "clarity" in that he seems to be forcefully arguing for positions about which there actually is some contention. (Though I have only read a fraction of it.)

It appears to me that most of the unclarity in philosophy is not of the sort that more intelligence can fix. Rather, it seems that most of philosophy is about matters of subjective opinion, or problems that simply aren't well-formulated, which cannot be solved because there actually is no objective solution. If it were otherwise, I would expect that philosophers would have resolved quite a lot of previously contentious issues, just as every science has done. So this is why I have pretty strong doubts that there will be a consensus about this particular problem.

@HarrisonNathan I think there are lots of problems that dogs think are insoluble that humans have solved and similarly I don't put much stock in humans thinking that problems are insoluble unless they have a proof.

For this specific problem I notice that we are living in a quantum universe where anthropics are a live issue, and there are likely facts about the most efficient way for an epistemic agent to reason and therefore how it should think about its own degree of belief.

Yes we need to bind the words to the facts and the maths but there is typically a best way to do that binding once a problem is resolved.

@MartinRandall Well, consider the question "what's the best color?" This is a question a small child might ask expecting someone wiser would know the right answer, but adults recognize it as a question without a conceivable answer.

You would like someone wiser to answer the question "what's the best way to reason?" It's not obvious to me that this is a different sort of question.

@HarrisonNathan there's surely a most aesthetically pleasing color averaged over all small children currently alive. Probably a pink of some kind.

Not a question that humans could conceivably answer.

@MartinRandall What you just did there is an example of making up a standard, and thereby transmuting the question into a different one.
That's exactly what philosophers do to questions such as "what's the best way to reason?" They formulate multiple entirely different adjacent questions, then call each other crazy. (This is unique to philosophy as far as I know - intelligent, skilled practitioners of the subject thinking not merely that their colleagues are wrong but that they are "crazy" and simply not seeing the obvious.)

@HarrisonNathan when the small child asked "what is the best color" they presumably meant something, and based on my small child experience they probably meant aesthetics-to-children. But of course if they meant something else there can be a different answer, or none.

@HarrisonNathan I see what you're getting at and I think you could make a prediction market on when we (broadly) will get to consensus on this question, if ever.

@MartinRandall They presumably didn't mean aesthetics averaged over all children - that's a bit advanced - and they certainly didn't mean to specify any of the various brain measurements one might do in order to operationalize that; neither did they think of the mathematical weightings that should apply to different levels of pleasantness of the colors in order to take the average. You can concoct all those things, and you can do it in a lot of different ways, but you will be answering questions that are different than the one the child asked.
Most of philosophy looks this way to me.
That's not to say nothing can ever be proven: indeed, some statements are tautologically false. However, I wouldn't have high confidence that any given controversial question is actually formulated well enough that a conceivable answer exists.

Proof of 1/2 by reductio ad absurdum:

Assume the thirder position is correct. Let's exaggerate the situation and say that you are woken up a million times if it's tails. Since you are a thirder, when you wake up you will be 999,999/1,000,000 confident that a tails was flipped. In plain English, we would say that you know that tails was flipped.

You can deduce the above before the experiment is run. That is, you know that you will know that tails was flipped. If you know that you will know X, then you you know X. Therefore, you know that tails will be flipped before the experiment begins. This is obviously wrong.

@ItsMe This is just the normal "no updating" argument for 1/2 but with different numbers. Read a few of the linked papers.

@ItsMe take it further.

If instead the rules say sleeping beauty will not be woken at all if is heads and will be woken up once if it is tails. They know before the experiment runs that if they are woken it was tails, because waking them up gives new information and they should update on that information.

@ItsMe It's a nice thought experiment. I agree with @MartinRandall and @ShitakiIntaki - you will know that tails was flipped on Tuesday. But you won't know that tails was flipped on Wednesday or Monday. You will even know on Monday that you will know that tails in flipped on Tuesday. But you still won't know that tails will be flipped on Monday.

@ItsMe I don't see how this is a reductio ad absurdum. It's not obviously wrong to me.

A wheel will be spun with numbers from 1 - 100,000 resulting with equal probability. If the number 50,000 results, you will be woken up 10 million times. On any other number, you will be woken up once.

Each time they wake you up, you will be asked what you think the probability is that the number was 50,000. After you state your probability, they will tell you the number.

There is no bet, but you are being monitored by a lie detector. If you are unsurprised by an outcome you claimed was very unlikely, or surprised by an outcome you claimed was very likely, you will be judged a liar.

"you can't just judge a mathematical question on subjective qualities like surprise!"

Yes I can, midwit. In fact there is no other way to do it; nothing in any math will ever tell you how to line itself up with material facts. It is up to you to do that, not math.

In a world where such situations were common, it is possible (depending on how those situations were related to other facts) that people's surprise would work differently.

That is not a problem; I just said that how you line up material facts with math is not a matter of math. In a world where putting two apples next to two others results in two of them merging and leaving three, we would line up "2 + 2 = 4" differently with material facts.

We don't live in either of those worlds. In the one we live in, two apples and two apples making 4 is a good example of 2 + 2 = 4 and Sleeping Beauty is a good example of a 50/50 probability.

@DavidBolin until the last sentence I thought this was a thirder argument.

I've finished my post about betting in Sleeping Beauty and two adjacent cases. I don't think there are any unaddressed arguments in favor of thirdism left?

bought Ṁ20 YES from 50% to 52%

@a07c Are you going to write this for publishing in a philosophy/math journal after getting feedback on LessWrong?

@MartinRandall yes, that's the plan.

@a07c I honestly think you've begged the question, this is still an impressive writeup though.

This situation is essentially making a bet on an outcome of a coin toss, and then the same bet has to be repeated if the coin comes Tails. Betting on 1:2 odds doesn't say anything about the unfairness of the coin or having some new knowledge about its state. Instead, it's fully described by the unfairness of the betting scheme which rewards Tails outcomes more.

This is the crux of the disagreement, I think. I believe, fully and exactly that credence and betting odds are the same thing. They can diverge, (there was a paper about this linked below) if you put your finger on the scale in some way, but in the original question betting on tails pays out iff the coin is actually tails, so I don't have an issue with treating them as the same in this case. I guess I'm waiting again, now that I've read your betting post, to read about the differences between probability, credence, and betting odds.

@a07c your post argues that if a tree falls on a deaf Sleeping Beauty, in a forest with no one to hear it, it surely doesn’t produce a sound, because here’s how humans perceive sounds, which is the definition of a sound, and there are no humans. (Or maybe that is surely produces the sound, because here’s the physics of the sound waves, and the tree surely abides by the laws of physics and there are sound waves.)

Y’all are arguing about definitions. You feel strongly that “probability” is that thing that triggers the “probability” concept neuron in your brain. If people have different concept triggering “this is probability”, you feel like they must be wrong, because they’re pointing at something they say is a sound and you say isn’t.

Probability is something defined in math by necessity. There’s only one way to do it to not get exploited in natural betting schemes that everyone accepts. But if there are multiple copies of the agent, there’s no longer a single possible betting scheme defining a single possible “probability”, and people draw the boundary/generalise differently in this situation

You all should just call these two probabilities two different words instead of arguing which one is the correct one.

(@MartinRandall the answer depends on the definitions, the market should resolve to 50% or N/A. Note that I haven’t traded on the market)

@ms I agree that we can give fair betting odds for any particular betting scheme, and we can give ideal predictions for any particular scoring scheme.

I didn't think there is yet a consensus of philosophers that "degree of belief" for an "ideally rational epistemic agent" is best understood in those terms.

A consequence of the "scoring scheme" definition is that Beauty's "degree of belief" can be "purple" in a pathological scoring scheme. That makes it unappealing to me.


You feel strongly that “probability” is that thing that triggers the “probability” concept neuron in your brain.

Probability has a mathematical definition.

@KongoLandwalker And the world is not math, so there is nothing in that definition telling you which things in the world to match up with that definition.

This is why the question exists in the first place.

E.g. lay out two apples and two more apples. Count them and you get 4.

Is this an example of 2 + 2 = 4?

Maybe, maybe not, but math will not tell you whether that is an appropriate example or not.

@DavidBolin when people want to specifically avoid math they use the word "credence" and calmly talk about their feelings. But trying to rewrite a good and precise definition of probability is a sabotage and addition of ambiguity into the debate.

Thirder solution does not satisfy criteria of mathematical probability. All outcomes have to be exclusive.

@DavidBolin with your example: yes, i would agree that 2+2equal 4. But imagine people without mathematical reasoning coming into math and trying to say 2+2=6, and instead of reading a book about +, they juggle with words "odds"/"credence"/"probability"/"anthropics" like they are interchangable and obfuscate the discussion by not learning how probability is defined.

When thirder says his credence or feel or something is 1/3, i do not care and do not correct him. But when a thirder says probability is 1/3, then he specifically violates math.

bought Ṁ10 NO


I honestly think you've begged the question

How so? I'm building based on a previous result, which, as you agreed, is necessary correct.

This is the crux of the disagreement, I think

I wouldn't call it a crux. I'm describing halfer narrative, for those who have troubles understanding it. Likewise previously I was saying that thirders treat per awakening bet as fair. This disagreement is simply about semantics. What is really important is that both reasonings agree on per experiment and per awakening betting schemes.

I believe, fully and exactly that credence and betting odds are the same thing.

Which ones? There are odds for per awakening bet and for per experiment bet and these odds are different. People usually say that whichever odds feel more natural to them are the ones that credence should be about, but then there is a disagreement about intuitions. So appealing to it can't resolve the issue. Also, obviously, it's not that math depends on what you or I find intuitive, it's the other way around. We should find intuitive what the math says and if we don't - it's a resin to modify our intuitions.

I guess I'm waiting again, now that I've read your betting post, to read about the differences between probability, credence, and betting odd

Betting odds depend on both probability and utility of the event, as explained in this post. That's why people who disagree on probabilities can still agree on betting odds. Credence is probability conditional on available evidence. And as I show in the previous post, there is only one sound way to define probability space in Sleeping Beauty. Both probability and credence that the coin is Heads on Beauty's awakening are 1/2. And being odds depend on whichever betting scheme is proposed.