Sleeping Beauty as a decision problem (solved)

by Bongo2 min read10th Oct 20103 comments

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Sleeping Beauty Paradox
Personal Blog

EDIT: User:Misha solved it


 

First, here's the Sleeping Beauty problem, from Wikipedia:

The paradox imagines that Sleeping Beauty volunteers to undergo the following experiment. On Sunday she is given a drug that sends her to sleep. A fair coin is then tossed just once in the course of the experiment to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a second dose of the sleeping drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday, without flipping the coin again. The sleeping drug induces a mild amnesia, so that she cannot remember any previous awakenings during the course of the experiment (if any). During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.


Each interview consists of one question, "What is your credence now for the proposition that our coin landed heads?"


I was looking at AlephNeil's old post about UDT and encountered this diagram depicting the Sleeping Beauty problem as a decision problem.



This diagram is underspecified, though. There are no specific payoffs in the boxes and it's not obvious what actions the arrows mean. So I tried to figure out some ways to transform the Sleeping Beauty problem into a concrete decision problem. I also made edited versions of AlephNeil's diagram for versions 1 and 2.


The gamemaster puts Sleeping Beauty to sleep on Sunday. He uses a sleeping drug that causes mild amnesia such that upon waking she won't be able to remember any previous awakenings that may have taken place during the course of the game. The gamemaster flips a coin. If heads, he wakes her up on monday only. If tails, he wakes her up on monday and tuesday.

Version 1

Upon each awakening, the gamemaster asks Sleeping Beauty to guess which way the coin landed. For each correct guess, she's awarded $1000 at the end of the game. diagram

Version 2

Upon each awakening, the gamemaster asks Sleeping Beauty to guess which way the coin landed. If she all of her guesses are correct, she's awarded $1000 at the end of the game. diagram

Version 3

Upon each awakening, the gamemaster asks Sleeping Beauty for her credence as to whether the coin landed heads. For each awakening, if the coin landed x, and she declares a credence of p that it landed x, she's awarded p*$1000 at the end of the game.

Version 4

Upon each awakening, the gamemaster asks Sleeping Beauty for her credence as to whether the coin landed heads. At the end of the game, her answers are averaged to a single probability p, and she's awarded p*$1000.


What's interesting is that while the suggested answers for the classic Sleeping Beauty problem are (1/2) and (1/3), for neither version 1 nor 2 is the correct answer to guess heads every second or third time, and for neither version 3 nor 4 is the correct answer to declare a credence of (1/2) or (1/3). The correct answers are (correct me if I'm wrong, I got these by looking at AlephNeil-style UDT diagrams and doing back-of-the-envelope calculations):

  • Version 1: Always guess tails. Expected payoff $1000
  • Version 2: Always guess heads, or always guess tails. Expected payoff $500.
  • Version 3: Answer with a 0% credence of heads. Expected payoff $1000.
  • Version 4: All answers seem to have an expected payoff of $500.

Is there any way to transform Sleeping Beauty into a decision problem such that the correct answer in some sense is either (1/2) or (1/3)?

Is there a general procedure for transforming problems about credence into decision problems?

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3 comments, sorted by Highlighting new comments since Today at 1:16 AM
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[-][anonymous]10y 16

Sleeping Beauty should believe in a 1/3 probability of heads, but this doesn't mean she should guess "heads" 1/3 of the time, and "tails" 2/3 of the time. It makes no sense to ever go with the option that has a smaller chance of winning, if the object is to correctly guess the coin toss as often as possible.

I haven't checked the math, but if I remember my information theory correctly, the proper way to elicit an accurate prediction from Sleeping Beauty is the following: ask her for a credence P that the coin is heads. If the coin is heads, she gets $1000 log(P). If the coin is tails, she gets $1000 log(1-P).

Since 0<P<1, she's losing money either way, so if that bothers you, pay her some constant amount of money every time you do this to make up for it.

P. S. See "Scoring rule" on Wikipedia for the more general case.

So to elicit honest credences, scale the payoff by the log of the credence. And the whole problem here was how to elicit honest credences from Sleeping Beauty. You just solved my whole problem, thanks!

Also, I think it's time for me to reread Technical Explanation.

By the way, this was part of a project to make easier edited versions of AlephNeil's diagrams to aid in understanding the post, but I got stuck on Sleeping Beauty.