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Suppse Sleeping beauty is told that there was some large number of coin flips, and that she is being woken on the nth flip as a result of it being heads or tails. Should that change her guess from if she was just told that the experiment was repeated once?

What if she was told she was woken from one of the n flips, but not necessarily the most recent one?

If we believe the Self-Indicating Assumption, I don't think the answer would change. But I'm not sure what would happen if we believed the Self-Sampling Assumption.

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1 Answer 1

As stated, this is not any different from the original problem: you've just added irrelevant stuff (that happens to be coin flips) as a premise in addition to the flip that actually determines whether she's awoken once or twice. Events that have no bearing on the outcome also have no bearing on the logic of the situation: adding irrelevant coin flips is no different than adding that she saw a pair of pigeons fly past the window after she woke up.

You might stop believing the Self-Sampling Assumption if it made you conclude otherwise, however.

Let me actually answer the Sleeping Beauty problem also, since writings on it seem confused, and there are apparently two different philosophical camps that each in their own Wikipedia summary fail to grasp the essence of the situation.

The situation is as follows (made to sound more like the fable): Sleeping Beauty goes to sleep; a coin is flipped. If it comes up heads, the Prince kisses her on the lips, she wakes up, and stays awake. If it comes up tails, the Prince kisses her, she wakes up; then the Prince pricks her finger again and she falls back asleep; and then he kisses her and she wakes up again, unaware that she was previously awake for a while. Immediately when she wakes up, in any case, we ask her: Heads or Tails! What should she say?

She should say, "Why are you bothering me with this question?!"

So, okay, Sleeping Beauty has recognized that if you want to maximize your score in a game, you need to know what the rules for scoring are. You don't need to know where the pigeons are flying or if other people are playing the same game as you, as long as the pigeons and other players do not affect your game. It does matter if the structure of the game introduces correlations in an otherwise random process.

Let's try some rules, then.

  1. You get a point every time you get the answer right. (In the tails case, you can win 2 points; in heads, only 1.) Solution: guess tails. 50% of the time you will get 0 points, 50% of the time you'll get 2, for a net payoff of 1 point. (If you guess heads, your expected payoff is 1/2.)

  2. You get a point if you are consistently right; half a point if you're right half the time; no points if you are never right. Solution: it doesn't matter what you guess. Expected payoff is 1/2 regardless of what you say.

So, if Sleeping Beauty wants to maximize number of questions answered correctly, she'll choose tails, while if she wants to maintain accurate thought for the maximal duration, she's free to choose randomly until she gets information. If you ask her to calculate probabilities instead of give a single answer, then it matters what exactly you ask.

If SIA vs. SSA boils down to anything other than asking different questions / scoring your rewards differently, then making that assumption is unjustified.

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Suppose 100 coin flips -> 50 people from heads, 100 from tails. From SSA, this means we should give 2/3 probability to tails, no? (I agree that the extra info is irrelevant if you believe SIA.) –  Xodarap Feb 2 '13 at 16:55
Revised my answer to clarify the issue. –  Rex Kerr Feb 2 '13 at 17:46
When you're looking at expected payoffs you're using SIA, no? (You might believe that that is the correct assumption to make, and I actually agree, but I'm curious what SSA would say.) –  Xodarap Feb 3 '13 at 15:25
@Xodarap - I am using standard probability theory without paying any attention to whether something happens to be an observer or not. I'm not, as the description does on Wikipedia, saying that there are "three observers". There is one, with three possible states in which said observer might be asked heads-or-tails, and 100% correlation between two of those states. You then reason as you would with any other probability problem. The prior "I am an observer" doesn't alter anything in this case. –  Rex Kerr Feb 3 '13 at 17:24

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