The Sleeping Beauty Paradox or the Computer Decision Tree Problem

The Sleeping Beauty Paradox grows out of the Absent-Minded Driver Paradox which in turn stems from a computer programming problem studied by M. Piccione and A. Rubenstein in the journal Games and Economic Behavior, 1997. We first consider the form of the problem ( rather than paradox) as stated in the so-called Sleeping Beauty Paradox. Here is the problem as originally posed by Jamie Dreier in an internet recreational puzzle discussion group:

• We plan to put Beauty to sleep by chemical means, and then we'll flip a fair coin, i.e. the probability of heads is 1/2 and the probability of tails is 1/2.
• If the coin lands heads, we will awaken Beauty on Monday afternoon and interview her. If it lands tails, we will awaken her Monday afternoon, interview her, put her back to sleep, and then awaken her again on Tuesday afternoon and interview her again.
• In each case the interview is to consist of the one question: what is your credence now for the proposition that the flipped coin landed heads?
• Under no circumstances will Beauty be able to tell which day it is, nor will she remember whether she has been awakened before. She only knows the details of the experiment.
• What credence should she state in answer to the interview question?

In the form of the Absent-Minded Driver one version allows that there consequences for poor choices. Hence, the problem involves what is called the maximization of the expected outcome. As stated, the Sleeping Beauty Problem asks the question: what credence should Sleeping Beauty give to the two outcomes of the coin toss?

This question does not make clear what is meant by credence. It appears that rather than restating the Absent-Minded Drivers Problem, the Sleeping Beauty problem is inspired by it and only closely related to it. Most interpretations of the Sleeping Beauty Problem give a solution in terms of probability theory and this seems to be a reasonable approach. Some interpretations appear to consider the question of credence to be related to expected correct answers.

How should the game be interpreted and what is the correct answer under the various interpretations?