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Comment by jamesrom | Hacker News Reader

Can someone explain the half position clearly? I don't understand how anyone could think half is the correct answer.

Let me frame the question a different way. You are one of three volunteers in separate rooms. I flip a coin and if it's heads I ask one volunteer (at random) to guess the outcome. If I flip tails I ask two of the volunteers (again, at random) to guess the outcome.

You know the rules I will follow, but you cannot tell if anyone else has been asked before you. I open the door and ask you to guess the outcome of the flip. What do you guess?

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Your reframing of the question changes the problem. Let A be the probability of Heads (1/2) and B be the probability you are asked (1/2) then P(B|A) is 1/3. So the probability the coin came up heads is P(A|B) = P(B|A) * P(A)/P(B) = 1/3. Which is what you expected.

In the original problem the probability of getting woken up is 1. So we end up with the probability of the coin being heads given that I was just woken up is (1 * 1/2) /1 = 1/2.

Suppose I am going to flip a coin and give you money if you guess correctly. On heads I hand you 1$ on tails I hand you 1$ and add 1$ to your bank account without telling you. Now, after flipping the coin I ask you what the heads or tails?

Clearly your going to pick tails. Why? Because it's the more rational choice based on the spoken rules.

Going to sleep and losing memory is really just window dressing. The simple truth is before you went to sleep you should be able to say 1/3 odds of waking up to heads.

But that's still not the question.

The stated question is When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?

Not, Should you _guess_ Heads or Tails?

The probability of the coin toss _does not change_. The probability that "the reason they are waking you up is because the coin was Heads" does change, but that's not the question.

Clearly there is a 50:50 shot you get heads or tails from the coin. In your model of it's a 50:50 coin then you would have 50:25:25. The problem is waking up the second time does not reduce the odds of waking up the first time thus 50:50:50.

I think you changed the question. Before you go to sleep you know you have a 50% chance of waking up to heads and a 50% chance of waking up to tails twice. When you wake up you haven't learned anything so those probabilities can't have changed.

You have arguably less information when you wake up in that you don't know the time. Are you waking up because it's tails? because it's heads? or is it tails the second time?

This is an incredibly powerful rephrasing, kudos. It's perfect, because for all intents and purposes you are "a[nother] person, who cannot tell if anyone [he] has been asked before", even though that other person is "also you."

Very good! The half position is clearly, simply wrong.

I think 1/2 argument makes essential use of the fact that the "2 people" are the same person at different times. Ultimately it comes down to a different idea of the way "being right twice" is valued. In the 1/3 position, the model is that each time you answer the question you get something good if you're right. In the 1/2 perspective, the idea is that after the whole experiment is over you either were simply correct or incorrect, and being correct twice doesn't count any extra.

You don't get to be "right twice"

The question is When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?

It's not a guessing competition, it's asking you to assess the probability of a past event that you have no new information about.

I like this rephrasing, but I think you only need n=2 subjects not n=3. To see why you should guess tails, set n=100.

The fact is, there aren't three volunteers, there's only one. Upon awakening, she has no information she didn't have when she went to sleep, which at first seems to be an unassailable argument in favor of the "halfers."

To me, the problem with the "no new information" argument is that there's another popular paradox in which it also seems to apply, but fails spectacularly. In the Monty Hall paradox (https://en.wikipedia.org/wiki/Monty_Hall_problem), you know in advance that the host is going to open a door at random and offer you a chance to change your initial guess. You're aware that the host knows what's behind the doors. You know that no matter what door you pick, one of the remaining ones has a goat behind it. And you know that the omniscient host will pick that door. So what new information do you gain when the host does what you knew he was going to do?

One way to resolve the Monty Hall paradox is to rephrase it such that there are a million doors and the host opens all of the remaining ones except one. Of course it makes sense to switch doors in that scenario, since you knew that the host would avoid opening your initially-chosen door as well as the one with the car behind it. Your initial door has 1:1000000 odds of winning, while the only other closed door has odds of 1/2.

So if you could find a way to transform the Sleeping Beauty problem into a Monty Hall problem, that could be a good way to settle the argument in favor of the "thirders."

Or, if with some logical sleight-of-hand, you could show that SB is not isomorphic to MH, that could potentially be used to strengthen the no-new-information argument made by the "halfer" side. I'm leaning that way, because SB does not know upon awakening, and will not remember, whether the experiment will result in her being awakened twice. For all she knows, she's playing the same Monty Hall game for the second time. It's this lack of an unbroken timeline of conscious awareness that settles the question in favor of the halfers.

To answer the thirders' rebuttal that SB would lose money if she were gambling on the coin toss with any odds other than 1:3, I'd say that's a completely different experiment. We aren't being asked to speculate on the set of events behind multiple awakenings, just one in isolation. The lack-of-new-information argument carries the day. Any arguments that involve repetition, clones, or gambling fail because they are responding to a different question.

TL,DR: Don't gamble if the game involves erasing the players' memories. You will lose a lot of money in a hurry, even if your strategy is mathematically optimal.

Here's an amplification that might clarify it. If you flip heads, I'll wake you up zero times. If you flip tails, I'll wake you up once. Now what's the probability that you flipped tails. 100%, right? Why? The coin still has a 50% chance of giving either heads or tails.

Here's another -- if you flip heads, I'll wake you up once. If you flip tails, I'll wake you up a million times. When I wake you up, is it more likely that you flipped tails or heads, remembering that you don't know which day it is.

You are right, by the way, that there is no new information -- the tendency to answer "50%" the night before is the logical error. You are conflating the odds of flipping heads or tails with the odds of WHETHER you flipped heads or tails in light of the known outcome (which is not 1:1 with flipping heads or tails).

Also, if you believe in statistics, the right answer in the continuous case (infinite repeats) is also the right answer in the singular case (one repeat). Your point about "there aren't three volunteers, only one" is fallacious.

I do like that, it's a strong argument. But it still relies on monkeying with the participant's awareness in a way that is likely to (or in this case, will) keep them from making the right decision over time. You use the term "known outcome," but there isn't a known outcome if you're dead or asleep forever.

Your point about "there aren't three volunteers, only one" is fallacious.

But so is treating it as any other problem (like gambling) that only converges on an answer after multiple trials. The strength of your argument is that it doesn't rely on answering a different question, just changing a parameter in the question that was asked. But it's an awfully important parameter. Implicit in the question is the premise that you will be alive/aware at some point in the future, so that you can be asked what you think.

Edit: another way to rebut your particular argument is by sticking with the original terms of the problem, except that if the coin lands on heads, they shoot you after you answer. That doesn't help you answer the question.

I think you're starting to grasp it. The key here is that if you are dead or asleep forever, then you are not waking up to answer the question. Therefore, the simple fact that you are waking up brings new information to the situation -- you know for sure that you didn't flip heads.

But in the original question, you were already guaranteed to wake up, so how does it give you any information?

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