The Sleeping Beauty problem, also known as the Sleeping Beauty paradox, is a puzzle in decision theory in which an ideally rational epistemic agent is told she will be awoken from sleep either once or twice according to the toss of a coin. Each time she will have no memory of whether she has been awoken before, and is asked what her degree of belief that "the outcome of the coin toss is Heads" ought to be when she is first awakened.
History The problem was originally formulated in unpublished work in the mid-1980s by Arnold Zuboff (the work was later published as "One Self: The Logic of Experience") followed by a paper by Adam Elga. 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. 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. A more recent paper by Peter Winkler discussing different sides of the problem was published in The American Mathematical Monthly in 2017.
The problem As originally published by Elga, the problem was:
Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads? There are three superficial differences between Zuboff's unpublished versions, and the one Elga actually solved (which is not quite the same as the one he asked). They should not affect the solution method. Zuboff used a large number, N, of days. There was to be one waking per day after an unspecified coin-flip result, and one waking on a random day in that interval after the other result. Elga fixed N at 2, named Tails as the result where there were to be two wakings, and placed the one waking after Heads on day 1. This has become the canonical form of 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 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 (and thus their probabilities must add to 1), the probability of each is then 1/3 by the previous two steps in the argument. An alternative argument is as follows: Credence can be viewed as the amount a rational risk-neutral bettor would wager if the payoff for being correct is 1 unit (the wager itself being lost either way). In the heads scenario, Sleeping Beauty would spend her wager amount one time, and receive 1 money for being correct. In the tails scenario, she would spend her wager amount twice, and receive nothing. Her expected value is therefore to gain 0.5 but also lose 1.5 times her wager, thus she should break even if her wager is 1/3.
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. 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. Philosophers such as Christopher Hitchcock have argued against the halfer position by arguing that Sleeping Beauty is subject to Dutch books if she assigns a credence of 1/2. It has been argued that halfers can avoid Dutch books by adopting evidential decision theory. However, Vincent Conitzer argues that halfers are still affected by Dutch books even after adopting evidential decision theory.
Double halfer position The double halfer position argues that both P(Heads) and P(Heads|Monday) equal 1/2. Mikaël Cozic, 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.
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