Sleeping Beauty problem

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

History

The problem was originally formulated in unpublished work in the mid 1980s by Arnold Zuboff (the work was later published as "One Self: The Logic of Experience")[1] followed by a paper by Adam Elga.[2] A formal analysis of the problem of belief formation in decision problems with imperfect recall was provided first by Michele Piccione and Ariel Rubinstein in their paper: "On the Interpretation of Decision Problems with Imperfect Recall" where the "paradox of the absent minded driver" was first introduced and the Sleeping Beauty problem discussed as Example 5.[3][4] The name "Sleeping Beauty" was given to the problem by Robert Stalnaker and was first used in extensive discussion in the Usenet newsgroup rec.puzzles in 1999.[5]

The problem

Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:

  • If the coin comes up heads, Sleeping Beauty will be awakened and interviewed on Monday only.
  • If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.

In either case, she will be awakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Sleeping Beauty is asked: "What is your credence now for the proposition that the coin landed heads?"

Solutions

This problem continues to produce ongoing debate.

Thirder position

The thirder position argues that the probability of heads is 1/3. Adam Elga argued for this position originally[2] as follows: Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. By even a highly restricted principle of indifference, given that the coin lands tails, her credence that it is Monday should equal her credence that it is Tuesday, since being in one situation would be subjectively indistinguishable from the other. In other words, P(Monday | Tails) = P(Tuesday | Tails), and thus

P(Tails and Tuesday) = P(Tails and Monday).

Suppose now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. Guided by the objective chance of heads landing being equal to the chance of tails landing, it should hold that P(Tails | Monday) = P(Heads | Monday), and thus

P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday).

Since these three outcomes are exhaustive and exclusive for one trial, the probability of each is one-third by the previous two steps in the argument.

Halfer position

David Lewis responded to Elga's paper with the position that Sleeping Beauty's credence that the coin landed heads should be 1/2.[6] Sleeping Beauty receives no new non-self-locating information throughout the experiment because she is told the details of the experiment. Since her credence before the experiment is P(Heads) = 1/2, she ought to continue to have a credence of P(Heads) = 1/2 since she gains no new relevant evidence when she wakes up during the experiment. This directly contradicts one of the thirder's premises, since it means P(Tails | Monday) = 1/3 and P(Heads | Monday) = 2/3.

Nick Bostrom argues that Sleeping Beauty does have new evidence about her future from Sunday: "that she is now in it," but does not know whether it is Monday or Tuesday, so the halfer argument fails.[7] In particular, she gains the information that it is not both Tuesday and the case that Heads was flipped.

Double halfer position

The double halfer position[8] argues that both P(Heads) and P(Heads | Monday) equal 1/2. Mikaël Cozic,[9] in particular, argues that context-sensitive propositions like "it is Monday" are in general problematic for conditionalization and proposes the use of an imaging rule instead, which supports the double halfer position.

Connections to other problems

Nick Bostrom argues that the thirder position is implied by the Self-Indication Assumption.

Credence about what precedes awakenings is a core question in connection with the anthropic principle.

Variations

Extreme Sleeping Beauty

This differs from the original in that there are one million and one wakings if tails comes up. It was formulated by Nick Bostrom.

The Sailor's Child problem

The Sailor's Child problem, introduced by Radford M. Neal, is somewhat similar. It involves a sailor who regularly sails between ports. On one port there is a woman who wants to have a child with him, across the sea there is another woman who also wants to have a child with him. The sailor cannot decide if he will have one or two children so he will leave it up to coin toss. If Heads he will have one child and if Tails two children. But if the coin lands on Head which woman would have his child? He would decide this by looking at The Sailors Guide to Ports and the woman in the port that appears first would be the woman that he has a child with. You are his child. You do not have a copy of The Sailors Guide to Ports. What is the probability that you are his only child, thus the coin landed on Heads (assume a fair coin)?[10]

See also

References
  1. Arnold Zuboff (1990). "One Self: The Logic of Experience". Inquiry: An Interdisciplinary Journal of Philosophy. 33 (1): 39–68. doi:10.1080/00201749008602210.(subscription required)
  2. Elga, A. (2000). "Self-locating Belief and the Sleeping Beauty Problem". Analysis. 60 (2): 143–147. CiteSeerX 10.1.1.32.3107. doi:10.1093/analys/60.2.143. JSTOR 3329167.
  3. Michele Piccione and Ariel Rubinstein (1997) “On the Interpretation of Decision Problems with Imperfect Recall,” Games and Economic Behavior 20, 3-24.
  4. Michele Piccione and Ariel Rubinstein (1997) “The Absent Minded Driver's Paradox: Synthesis and Responses,” Games and Economic Behavior 20, 121-130.
  5. Nick Wedd (June 14, 2006). "Some "Sleeping Beauty" postings". Retrieved November 7, 2014.
  6. Lewis, D. (2001). "Sleeping Beauty: reply to Elga" (PDF). Analysis. 61 (3): 171–76. doi:10.1093/analys/61.3.171. JSTOR 3329230.
  7. Bostrom, Nick (July 2007). "Sleeping beauty and self-location: A hybrid model" (PDF). Synthese. 157 (1): 59–78. doi:10.1007/s11229-006-9010-7. JSTOR 27653543.
  8. Meacham, C. J. (2008). "Sleeping beauty and the dynamics of de se beliefs". Philosophical Studies. 138 (2): 245–269. CiteSeerX 10.1.1.517.4904. doi:10.1007/s11098-006-9036-1. JSTOR 40208872.
  9. Mikaël Cozic (February 2011). "Imaging and Sleeping Beauty: A case for double-halfers". International Journal of Approximate Reasoning. 52 (2): 137–143. doi:10.1016/j.ijar.2009.06.010.
  10. Neal, Radford M. (2006). "Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning". arXiv:math/0608592.

Other works discussing the Sleeping Beauty problem
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