# Sleeping Beauty problem

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The problem was originally formulated in unpublished work in the mid 1980s by Arnold Zuboff (work that was later published as "One Self: The Logic of Experience"[1]), followed by a paper by Adam Elga.[2] A formal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, 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, 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 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, 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).

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

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.

## Operationalization

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. If we include Tuesday responses, the answer is 1/3.

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 (500 heads and 1000 tails, on average). 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 want to estimate, not the statistics of the circumstances of her awakenings, but rather the statistics of coin tosses that precede all awakenings. Such statistics would give the answer P(Heads) = 1/2, but it would be impossible for Beauty to collect these statistics at the times of her awakenings.

It's even simpler with bets: If Beauty and the experimenter agree that bets from all awakenings are counted, 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 - but that Beauty is no longer asked to commit to a bet, but rather to say how she would bet if it were Monday.

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.[citation needed]

## 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

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 M. Neal.[10] 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

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

## References

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. ^ 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". 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. 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. ^ https://arxiv.org/pdf/math/0608592.pdf