Sleeping Beauty problem

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The problem was originally formulated in unpublished work in the mid-1980s by Arnold Zuboff (the work was later published as "One Self: The Logic of Experience") followed by a paper by Adam Elga.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?