The Sleeping Beauty Problem

Author: Daniel Peterson
Categories: Epistemology, Philosophy of Science, Logic and Reasoning
Word count: 998

Sleeping Beauty
Sleeping Beauty

Imagine that Beauty takes part in an experiment: on Sunday night, she is put to sleep. Then, the experimenters flip a fair coin. If the coin lands heads, Beauty is awakened on Monday, then is put back to sleep until the experiment ends on Wednesday. If the coin lands tails, Beauty is awakened on both Monday and Tuesday; however, after her Monday waking, Beauty is given a memory drug that makes her forget her Monday waking when she wakes up on Tuesday.

This chart sums up the story:

Sunday Monday Tuesday Wednesday
Heads Experiment begins; Beauty goes to sleep Beauty is awakened, then a few minutes later, put back to sleep Beauty is not awakened. Experiment ends; Beauty wakes up
Tails Experiment begins; Beauty goes to sleep Beauty is awakened, then a few minutes later, put back to sleep (after receiving memory drug) Beauty is awakened, then a few minutes later, put back to sleep Experiment ends; Beauty wakes up

This case sets up what’s called the “Sleeping Beauty Problem.” Its question is: What degree of belief, or credence, should Beauty assign to the claim “The coin landed heads” (call this H) when she awakens?

Credences are numbers from 0 to 1 that represent how strongly we believe a claim to be true. A credence of 1 represents complete certainty that a claim is true, a credence of 0 represents complete certainty that a claim is false, and a credence of 0.5 represents complete neutrality.

Answers to the question of what Beauty’s credence in “The coin landed heads” (H) ought to be are typically divided into two camps:

  • “halfers” say Beauty should assign H a credence of 1/2.
  • “thirders” say Beauty should assign H a credence of 1/3.

The “problem” in the Sleeping Beauty problem is that it’s not clear which of these two solutions is correct or why. Debates over this problem are relevant to epistemology, the philosophy of science, the philosophy of probability, and more.

1. The Halfer Argument

Before Beauty goes to sleep, halfers say, Beauty knows that the probability of the coin landing heads is 1/2, and she gains (and loses) no information between going to sleep on Sunday and waking up. Halfers conclude that she should assign a credence of 1/2 to H.[1]

This typical halfer argument jointly employs two principles of rationality: the Reflection Principle[2] and the Principal Principle.[3]

The Reflection Principle states that, if I know that I will have credence c in a certain claim tomorrow (without learning any new information in the meantime), then I should have credence c in that claim today. The Principal Principle states that our degrees of belief ought to line up with real-world probabilities.[4] When used together, these principles tell us that Beauty should have credence 1/2 in H on Sunday night (per the Principal Principle), and, since she learns no new information when she wakes, she should have that same credence (1/2) whenever she wakes.

2. Thirder Argument: the Principle of Indifference

The Principle of Indifference states that, when we have n different possibilities and no reason to expect any one of these possibilities over any other, we should assign a credence of 1/n to the claim that any one possibility will occur. Think about a fair die: we have no evidence that any one of the six identical die faces is more likely to result from a die roll. Thus, we should assign a credence of 1/6 to the claim “the die will land six on the next roll.”

The Principle of Indifference at first seems to favor the halfer since we have no reason to believe that the coin is any more likely to land heads than tails or vice-versa. But, thirders argue, what we should really be indifferent over is Beauty’s three indistinguishable possible waking events (waking Monday when the coin landed heads, waking Monday when the coin landed tails, and waking Tuesday when the coin landed tails), and so we should assign a credence of 1/3 to H.[5] The thirder’s way of dividing up the world into indistinguishable states is more detailed and fine-grained than the halfer’s and so, they argue, represents a better application of the Principle of Indifference.[6]

3. Thirder Argument: New Information

For those unconvinced by the previous argument, here’s another one: imagine that, instead of waking up twice if the coin lands tails, Beauty instead wakes up a hundred times (and receives the memory drug after each waking).

In this variant,[7] it’s easier to believe that Beauty gains some new, relevant information when she wakes: “I’m awake now,” information she seems a lot more likely to receive if the coin landed tails instead of heads.[8] Even if Beauty doesn’t learn new information about the coin or the world outside of herself when she wakes, she does learn new information about her place in the world (that is, self-locating information) that justifies her change in credence. If this justification is correct, it applies in our original case too since, once again, Beauty should expect herself to wake up more frequently if the coin lands tails and so should take the new information that she is awake as evidence that the coin landed tails.[9]

4. Conclusion

The Sleeping Beauty problem is a seemingly simple puzzle whose solutions have required philosophers to refine their concepts and principles of rationality in an attempt to better understand how and when our credences ought to change. Though most philosophers agree that the thirder’s position is correct, there is currently no consensus on what the best argument for that position is.

While interesting in its own right, the problem challenges us to answer some of the most fundamental questions in epistemology: Under what conditions are we rationally permitted (or required) to change our credences? And what counts as “new evidence” that should lead us to change our credences? Answers to these questions matter not just to the Sleeping Beauty problem but for a whole host of philosophical puzzles in philosophy of religion, philosophy of science, and rational choice theory.[10] [11]

Notes

[1] See Lewis (2001) for a more detailed presentation of this argument.

[2] The Reflection Principle was originally proposed in van Fraassen (1984) and van Fraassen (1995).

[3] The Principal Principle was originally proposed in Lewis (1980).

[4] An example of the Principal Principle in action: imagine that you about to roll a fair die and assign a credence to the claim “The next die roll will land on 6”. The principal principle says that, since the real-world, objective probability of this claim being true is 1/6, that’s the credence you should assign to this claim.

[5] See Elga (2000) for a more detailed presentation of this argument.

[6] To see why we should prefer utilizing more “fine-grained” approaches, consider “rolling above a 4” and “rolling a 4 or lower” on a fair die. I may have no knowledge of what the die’s result will be and so be indifferent between these options, which would lead me to improperly assign a credence of 1/2 to my rolling a 5 or a 6 on a fair die. But the problem with applying the principle of indifference this way is that I have more information about dice than I’m letting on, and I know that I can divide “rolling above a 4” on the die into two options (rolling a 5 or a 6) which are each indistinguishable from each other and from the other four options available (rolling a 1, a 2, a 3, or a 4). Since “rolling above a 4” corresponds to 2 of these 6 indistinguishable fine-grained options, I should assign this result a credence of 1/3 instead of 1/2.

[7] Another variant case worth considering comes from Dorr (2002).

[8] Halfers respond that waking up shouldn’t count as “new information” since Beauty knows before she goes to sleep that she will receive that information in the future since she wakes up at least once no matter what side the coin lands on.

[9] There are other prominent thirder arguments in the literature worth mentioning which have been omitted here in the interest of space. Some of the most important (and technical) use Dutch Book arguments, which reveal sets of irrational credences by showing that an agent with these credences would be led to accept a series of bets which, jointly, would be guaranteed to lose her money. Hitchcock (2004) and Briggs (2010) present important examples of this style of argument applied to the Sleeping Beauty problem.

[10] These include fine-tuning arguments for the existence of an intelligent designer or a multiverse (see Thomas Metcalf’s The Fine-Tuning Argument for the Existence of God) and the doomsday argument, which purports to show that our being alive at this particular point in human history serves as good evidence that the human species is unlikely to survive much longer. See Huemer (2018) for more on the connections between these puzzles and the Sleeping Beauty problem.

[11] Many thanks to Chelsea Haramia, Dan Lowe, Thomas Metcalf, and Nathan Nobis for their helpful comments on this article.

References

Arntzenius, Frank. 2002. “Reflections on Sleeping Beauty.” Analysis. 62(1). 53-62.

Briggs, Rachael. 2010. “Putting a value on Beauty.” Oxford Studies in Epistemology. 3. 3-34.

Dorr, Cian. 2002. “Sleeping Beauty: in defence of Elga.” Analysis. 62(4). 292-296.

Draper, Kai and Joel Pust. 2008. “Diachronic Dutch Books and Sleeping Beauty.” Synthese. 164(2). 281-287.

Elga, Adam. 2000. “Self-Locating belief and the Sleeping Beauty problem.” Analysis. 60(2). 143-147.

Gibbard Allan and William Harper. 1981. “Counterfactuals and two kinds of expected utility.” in Paradoxes of Rationality and Cooperation: Prisoner’s Dilemma and Newcomb’s Problem. eds. Richard Campbell and Lanning Sowden. 133-158.

Hitchcock, Christopher. 2004. “Beauty and the Bets.” Synthese. 139. 405-420.

Huemer, Michael. 2018. “Self-Locating Belief.” Paradox Lost: Logical Solutions to Ten Puzzles of Philosophy. 219-243.

Lewis, David. 1980. “A Subjectivist’s Guide to Objective Chance.” Studies in Inductive Logic and Probability vol. 2. eds. Rudolph Carnap and R.C. Jeffrey. 263-293.

Lewis, David. 2001. “Sleeping Beauty: reply to Elga.” Analysis. 61(3). 171-176

van Fraassen, Bas. 1981. “A Problem for Relative Information Minimizers in Probability Kinematics.” The British Journal for the Philosophy of Science. 32(4). 375–379.

van Fraassen, Bas. 1984. “Belief and the Will.” Journal of Philosophy. 81. 235-256.

van Fraassen, Bas. 1995. “Belief and the problem of Ulysses and the Sirens.” Philosophical Studies. 77. 7-37

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About the Author

Dan Peterson is a part-time professor of philosophy at Morehouse College and South Georgia State College. He received his Ph.D. from the University of Michigan in 2013 and specializes in the philosophy of physics, philosophy of science, and formal epistemology. He has research and teaching interests in metaphysics, philosophy of religion, philosophy of education, and ethics. He is also the co-founder and executive director of Mind Bubble, an educational nonprofit in Atlanta that provides local students with free tutoring and educational workshops. DanielJamesPeterson.com

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