Sleeping Beauty problem

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The Sleeping Beauty problem is a puzzle in decision theory in which an ideally rational epistemic agent is to be woken once or twice according to the toss of a coin, and asked her degree of belief for the coin having come up heads.

The problem was originally formulated in unpublished work in the mid 1980s by Arnold Zuboff (work that was later published as "One Self: The Logic of Experience"[1]), followed by a paper by Adam Elga[2] and is related to problems of imperfect recall such as the "paradox of the absent minded driver". 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.[3]

The problem[edit]


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


This problem continues to produce ongoing debate.

Thirder position[edit]

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, 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


Consider now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. She knows the experimental procedure doesn't require the coin to actually be tossed until Tuesday morning, as the result only affects what happens after the Monday interview. Guided by the objective chance of heads landing being equal to the chance of tails landing, it should therefore hold that P(Tails | Monday) = P(Heads | Monday), and thus


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[edit]

David Lewis responded to Elga's paper with the position that Sleeping Beauty's credence that the coin landed heads should be 1/2.[4] 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.[5] In particular, she gains the information that it is not both Tuesday and the case that Heads was flipped.

Double Halfer position[edit]

The double halfer position[6] argues that both P(Heads) and P(Heads | Monday) equal 1/2. Mikal Cozic,[7] 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.


The Sleeping Beauty puzzle reduces to an easy and uncontroversial probability theory problem as soon as we agree on an objective procedure how to assess whether Beauty's subjective credence is correct. Such an operationalization can be done in different ways: By offering Beauty a bet; more elaborately by setting up a Dutch book; or by repeating the experiment many times and collecting statistics. For any such protocol, the outcome depends on how Beauty's Monday responses and her Tuesday responses are combined.

Consider long-run average outcomes. Suppose the experiment were repeated 1,000 times. It is expected that there would be about 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday.

  • If Beauty herself collects statistics about the coin tosses (in a way that is not obstructed by memory erasure when she is put back to sleep), she would register one-third of heads. If this long-run average should equal her credence, then she should answer P(Heads) = 1/3.
  • However, being fully aware about the experimental protocol and its implications, Beauty may reason that she is not requested to estimate a statistics of the circumstances of her awakenings, but a statistics of coin tosses that precede all awakenings. She would therefore answer P(Heads) = 1/2.

It's even simpler with bets: If Beauty and the experimenter agree that bets from her different awakenings are cumulative, then a heads quota of 1/3 would be fair. If on the other hand Tuesday bets are to be discarded (being dummy bets, undertaken only to keep Monday and Tuesday awakenings indistinguishable for Beauty), then the fair quota would be 1/2.

All this seems to be consensual among philosophers. Therefore the Sleeping Beauty problem is not about mathematical probability theory. Rather, the question is whether subjective probability or credence are well-defined concepts, and how they must be operationalized.

Connections to other problems[edit]

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.


The days of the week are irrelevant, but are included because they are used in some expositions. A non-fantastical variation called The Sailor's Child has been introduced by Radford Neal. The problem is sometimes discussed in cosmology as an analogue of questions about the number of observers in various cosmological models.

The problem does not necessarily need to involve a fictional situation. For example, computers can be programmed to act as Sleeping Beauty and not know when they are being run; consider a program that is run twice after tails is flipped and once after heads is flipped.

Extreme Sleeping Beauty[edit]

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

See also[edit]


  1. ^ Arnold Zuboff (1990). "One Self: The Logic of Experience" (PDF). Inquiry: An Interdisciplinary Journal of Philosophy. 33 (1): 39–68. doi:10.1080/00201749008602210. Retrieved November 7, 2014. (subscription required)
  2. ^ a b Elga, A. (2000). "Self-locating Belief and the Sleeping Beauty Problem". Analysis. 60 (2): 143–147. doi:10.1093/analys/60.2.143. JSTOR 3329167. 
  3. ^ Nick Wedd (June 14, 2006). "Some "Sleeping Beauty" postings". Retrieved November 7, 2014. 
  4. ^ Lewis, D. (2001). "Sleeping Beauty: reply to Elga". Analysis. 61 (3): 171–76. doi:10.1093/analys/61.3.171. JSTOR 3329230. 
  5. ^ 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. 
  6. ^ Meacham, C. J. (2008). "Sleeping beauty and the dynamics of de se beliefs". Philosophical Studies. 138 (2): 245–269. doi:10.1007/s11098-006-9036-1. JSTOR 40208872. 
  7. ^ 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. 

Other works discussing the Sleeping Beauty problem[edit]

External links[edit]