Have you ever wondered what it would be like to wake up and not know what day it is or whether you have been awake before? This is not just a hypothetical scenario for people who suffer from amnesia or insomnia; it is also a philosophical puzzle that has baffled many thinkers for decades. The puzzle is known as the Sleeping Beauty Problem, and it challenges our intuitions about probability and self-locating beliefs.

The Sleeping Beauty Problem involves a hypothetical experiment in which a person, called Sleeping Beauty, is put to sleep on Sunday and then awakened once or twice depending on the outcome of a fair coin toss. If the coin lands heads, she is awakened only on Monday; if it lands tails, she is awakened on both Monday and Tuesday. In either case, she is given a drug that erases her memory of any previous awakenings. Each time she is awakened, she is asked to assign a probability to the proposition that the coin landed heads.

Sleeping Beauty Paradox — Whose fault is it really?

The problem arises from the fact that there are two seemingly plausible answers to this question, but they are incompatible. One answer is that Sleeping Beauty should assign a probability of 1/2 to heads, since that is the objective chance of the coin toss and she has learned nothing new since Sunday. This is called the halfer position. The other answer is that Sleeping Beauty should assign a probability of 1/3 to heads, since that is the relative frequency of heads among her possible awakenings. This is called the thirder position.

The Sleeping Beauty Problem has generated a lot of debate among philosophers, mathematicians, statisticians and computer scientists. Some have argued for one position or the other based on various principles of rationality, indifference, self-sampling, betting or quantum mechanics. Some have suggested that the problem reveals a flaw in our standard notions of probability or belief updating. Some have even claimed that Sleeping Beauty’s epistemic state cannot be captured by a single number and that she suffers from a cognitive malfunction.

The Sleeping Beauty Problem is not just a theoretical curiosity; it has practical implications for real-world scenarios where we have to reason about situations that involve imperfect recall, multiple copies or branching worlds. For example, some have applied the problem to analyze the Monty Hall problem, the Doomsday argument, anthropic reasoning, quantum suicide, cloning, time travel and artificial intelligence. The problem also raises interesting questions about how we should model agents who have self-locating beliefs and how we should evaluate their decisions and actions.

In this article, we will explore some of these applications and see how the Sleeping Beauty Problem can shed light on some of the most fascinating and challenging problems in philosophy and science.

The history and origin of the Sleeping Beauty Problem

The Sleeping Beauty 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.

The arguments and counterarguments for the halfer and thirder positions

The halfer position argues that Sleeping Beauty should assign a probability of 1/2 to heads, since that is the objective chance of the coin toss and she has learned nothing new since Sunday. The main argument for this position is based on the principle of conditionalization, which says that one should update one’s beliefs by conditioning on new evidence. Since Sleeping Beauty has no new evidence when she wakes up, she should not change her prior belief that heads has a probability of 1/2.

The main counterargument for this position is based on the principle of reflection, which says that one’s current beliefs should match one’s expected future beliefs. Since Sleeping Beauty knows that if she wakes up and learns that it is Monday, she will assign a probability of 1/3 to heads (by symmetry), she should already assign a probability of 1/3 to heads before she goes to sleep on Sunday. Otherwise, she would be violating the principle of reflection and changing her beliefs without any new evidence. This argument assumes that Sleeping Beauty can anticipate her future beliefs based on her current information and reasoning. However, some critics of this argument have challenged this assumption and argued that Sleeping Beauty cannot reliably predict her future beliefs in this scenario, due to the effects of the memory drug and the uncertainty about the day.

Photo by Shaojie on Unsplash

The implications of the paradox for Bayesian updating and conditionalization

The Sleeping Beauty Problem poses a challenge for Bayesian updating, which is a method of revising one’s beliefs in light of new evidence. According to Bayesian updating, one should multiply one’s prior probability by a likelihood ratio to obtain one’s posterior probability. The likelihood ratio is the probability of the evidence given the hypothesis divided by the probability of the evidence given the negation of the hypothesis. For example, if one’s prior probability of heads is 1/2, and one learns that it is Monday, then one should multiply 1/2 by the likelihood ratio of Monday given heads over Monday given tails. This gives 1/2 x (1/1) / (1/2) = 1/2 as the posterior probability of heads.

However, this method seems to fail in the Sleeping Beauty Problem, because it depends on how one defines the evidence. If one defines the evidence as “I am awake”, then the likelihood ratio is 1, since being awake is equally likely under heads and tails. This gives 1/2 as the posterior probability of heads, which agrees with the halfer position. But if one defines the evidence as “I am awake for the first time”, then the likelihood ratio is 2, since being awake for the first time is twice as likely under heads than under tails. This gives 1/2 x 2 = 2/3 as the posterior probability of heads, which agrees with the thirder position.

The problem is that Sleeping Beauty does not know whether she is awake for the first time or not, since she has no memory of any previous awakenings. So how should she define her evidence? Some have suggested that she should use a weighted average of both possibilities, based on her subjective probability of being awake for the first time or not. This leads to a posterior probability of heads that is between 1/2 and 2/3, depending on how confident Sleeping Beauty is about her temporal location. Others have suggested that she should use a different method of updating that does not rely on conditionalization, such as Jeffrey conditionalization or imaging.

The relation between the paradox and the principle of indifference

The principle of indifference is a principle of rationality that says that one should assign equal probabilities to events or hypotheses that are equally likely or indistinguishable. For example, if one has no reason to favor any particular side of a fair die, then one should assign a probability of 1/6 to each side. The principle of indifference seems to support both the halfer and thirder positions in different ways.

The halfer position can appeal to the principle of indifference by saying that Sleeping Beauty should assign equal probabilities to heads and tails, since they are equally likely according to the objective chance of the coin toss. The thirder position can appeal to the principle of indifference by saying that Sleeping Beauty should assign equal probabilities to each possible awakening, since they are indistinguishable from her perspective. The problem is that these two applications of the principle of indifference lead to contradictory results.

The principle of indifference has been criticized for being vague, arbitrary and inconsistent in various cases. For example, it is not clear how to apply it when there are infinitely many possibilities or when there are different ways of partitioning or describing

The connection between the paradox and self-sampling and self-indication assumptions

The Sleeping Beauty Problem also relates to two assumptions that are often used in reasoning about situations where one’s existence or observation is contingent on some hypothesis. These are called the self-sampling assumption (SSA) and the self-indication assumption (SIA).

The SSA says that one should reason as if one were a random sample from the set of all observers in one’s reference class. For example, if one knows that there are 10 billion humans alive, and that 5 billion of them are male and 5 billion are female, then one should assign a probability of 1/2 to being male, regardless of any other information.

The SIA says that one should reason as if one’s existence or observation provides evidence for hypotheses that imply more observers or observations. For example, if one knows that there are two possible worlds, one with 10 billion humans and one with 20 billion humans, and that both worlds are otherwise identical, then one should assign a higher probability to being in the world with 20 billion humans, since that world implies more observers.

The SSA and the SIA can be applied to the Sleeping Beauty Problem in different ways. The SSA seems to support the halfer position, since Sleeping Beauty can regard herself as a random sample from the set of all awakenings in the experiment, which has a probability of 1/2 of being heads. The SIA seems to support the thirder position, since Sleeping Beauty can regard her awakening as evidence for tails, since tails implies more awakenings.

However, both assumptions have been challenged for leading to paradoxes or counterintuitive results in other cases. For example, the SSA leads to the Doomsday argument, which says that we should assign a high probability to human extinction in the near future, based on our low birth rank among all humans who will ever live. The SIA leads to the Presumptuous Philosopher problem, which says that we should assign a high probability to our universe being extremely large and complex, based on our mere existence.

The application of the paradox to the Monty Hall problem and its variants

Photo by Sophie Dale on Unsplash

The Monty Hall problem is a famous puzzle in probability theory that involves a game show where a contestant has to choose one of three doors, behind one of which is a car and behind the other two are goats. After the contestant chooses a door, the host opens another door that reveals a goat, and then offers the contestant a chance to switch doors. The question is: should the contestant switch or stick with their original choice?

The answer is that the contestant should switch, since switching gives them a probability of 2/3 of winning the car, while sticking gives them a probability of 1/3. This is because by opening a door with a goat, the host effectively eliminates one of the wrong choices and transfers its probability to the other unchosen door.

The Monty Hall problem can be seen as a variant of the Sleeping Beauty Problem, where instead of being awakened once or twice depending on a coin toss, the contestant is shown one or two goats depending on their choice of door. The analogy is as follows:

• Choosing heads corresponds to choosing door 1.
• Choosing tails corresponds to choosing door 2 or 3.
• Being awakened on Monday corresponds to being shown goat A behind door 2 or 3.
• Being awakened on Tuesday corresponds to being shown goat B behind door 2 or 3.
• Being offered to switch doors corresponds to being asked about heads.

In this analogy, switching doors is equivalent to assigning a probability of 1/3 to heads (or door 1), while sticking with the original choice is equivalent to assigning a probability of 1/2 to heads (or door 1).

The application of the paradox to the Doomsday argument and anthropic reasoning

Photo by Kym MacKinnon on Unsplash

The Doomsday argument is an argument that uses probabilistic reasoning to estimate the future duration of human civilization. It is based on the idea that one’s birth rank among all humans who will ever live is a random sample from a uniform distribution. For example, if one knows that there are 10 billion humans alive today, and that one’s birth rank is 5 billion, then one should assign a probability of 1/2 to being in the first half of all humans who will ever live.

The Doomsday argument then uses this probability to infer that there is a 50% chance that the total number of humans who will ever live is less than twice the current population, which implies that human extinction is likely to occur in the near future. The argument can be generalized to other reference classes, such as observers in the universe or intelligent beings in the multiverse.

The Doomsday argument can be seen as a variant of the Sleeping Beauty Problem, where instead of being awakened once or twice depending on a coin toss, one is born once or twice depending on the duration of human civilization. The analogy is as follows:

• Choosing heads corresponds to being born in a short-lived civilization.
• Choosing tails corresponds to being born in a long-lived civilization.
• Being awakened on Monday corresponds to being born early in one’s civilization.
• Being awakened on Tuesday corresponds to being born late in one’s civilization.
• Being asked about heads corresponds to being asked about one’s civilization’s duration.

In this analogy, assigning a probability of 1/3 to heads (or a short-lived civilization) is equivalent to adopting the thirder position, while assigning a probability of 1/2 to heads (or a short-lived civilization) is equivalent to adopting the halfer position. Thus, the Doomsday argument is analogous to adopting the thirder position in the Sleeping Beauty Problem.

The Doomsday argument is an example of anthropic reasoning, which is a type of reasoning that takes into account one’s own existence or observation as evidence for some hypothesis. Anthropic reasoning can be applied to various scenarios where one’s existence or observation depends on some contingent factor, such as the fine-tuning of physical constants, the origin of life, or the simulation hypothesis. Anthropic reasoning often involves using the SSA or the SIA, which can lead to different conclusions depending on how one defines one’s reference class.

The application of the paradox to quantum mechanics and the many-worlds interpretation

Quantum mechanics is a branch of physics that describes the behavior of subatomic particles and phenomena. Quantum mechanics is based on the idea that physical systems can exist in superpositions of states, which are combinations of possible outcomes. For example, an electron can exist in a superposition of spin up and spin down states, which means that it has both spin values at once until it is measured.

Quantum mechanics also implies that physical systems can undergo quantum interference, which means that they can interact with each other in ways that depend on their superposition states. For example, two electrons can interfere constructively or destructively depending on their relative spin states, which results in different probabilities of detection.

Quantum mechanics poses a problem for our classical understanding of reality and measurement. According to quantum mechanics, when we measure a system in a superposition state, we observe only one of the possible outcomes, and the system collapses into that state. However, it is not clear how or why this collapse occurs, or what determines which outcome we observe. This is known as the measurement problem.

One possible solution to the measurement problem is the many-worlds interpretation (MWI), which says that there is no collapse of the wave function, but rather every possible outcome of a measurement actually occurs in a separate branch of reality. According to MWI, when we measure a system in a superposition state, we split into multiple copies along with the system, each copy inhabiting a different branch or world. In each branch, we observe one of the possible outcomes of the measurement, and we are unaware of the existence of the other branches or copies. This way, MWI avoids the need for a special role for the observer or a mysterious process of wave function collapse. However, MWI also raises many questions and difficulties of its own, such as how to define and count the branches, how to assign probabilities to the outcomes, and how to explain why we do not experience any interference or communication between the branches.

The application of the paradox to cloning and personal identity

Cloning is a process of creating genetically identical copies of an organism or a cell. Cloning can raise various ethical, legal and social issues, but it can also pose philosophical puzzles about personal identity and self-locating beliefs. For example, suppose that Alice is cloned into two identical copies, Bob and Carol, who are then placed in separate rooms. Bob and Carol have the same memories and personality as Alice, but they do not know that they are clones or that the other clone exists. If Bob or Carol is asked to assign a probability to being Bob or Carol, what should they say?

This scenario can be seen as a variant of the Sleeping Beauty Problem, where instead of being awakened once or twice depending on a coin toss, one is cloned once or twice depending on a random event. The analogy is as follows:

• Choosing heads corresponds to being cloned once.
• Choosing tails corresponds to being cloned twice.
• Being awakened on Monday corresponds to being Bob.
• Being awakened on Tuesday corresponds to being Carol.
• Being asked about heads corresponds to being asked about being cloned.

In this analogy, assigning a probability of 1/3 to heads (or being cloned once) is equivalent to adopting the thirder position, while assigning a probability of 1/2 to heads (or being cloned once) is equivalent to adopting the halfer position. Thus, the thirder position implies that Bob and Carol should assign equal probabilities to being Bob or Carol, while the halfer position implies that Bob and Carol should assign higher probabilities to being themselves than being the other clone.

The cloning scenario also raises questions about what makes a person the same or different from another person. Is personal identity determined by physical continuity, psychological continuity, or something else? How does cloning affect one’s moral rights and responsibilities? How should clones relate to their originals and to each other?

The application of the paradox to time travel and causal loops

Time travel is a hypothetical phenomenon that involves moving backward or forward in time. Time travel can create various paradoxes and logical puzzles, such as the grandfather paradox, which says that if one travels back in time and kills one’s grandfather before one’s father is born, then one would never exist and thus could not travel back in time. Another type of paradox involves causal loops, which are sequences of events that cause themselves. For example, suppose that Alice travels back in time and gives a book to Bob, who then writes the book and gives it to Alice in the future.

Causal loops can be seen as variants of the Sleeping Beauty Problem, where instead of being awakened once or twice depending on a coin toss, one experiences an event once or twice depending on a causal loop. The analogy is as follows:

• Choosing heads corresponds to experiencing an event once.
• Choosing tails corresponds to experiencing an event twice.
• Being awakened on Monday corresponds to experiencing an event before the loop.
• Being awakened on Tuesday corresponds to experiencing an event after the loop.
• Being asked about heads corresponds to being asked about experiencing an event once.

In this analogy, assigning a probability of 1/3 to heads (or experiencing an event once) is equivalent to adopting the thirder position, while assigning a probability of 1/2 to heads (or experiencing an event once) is equivalent to adopting the halfer position. Thus, the thirder position implies that one should assign equal probabilities to experiencing an event before or after the loop, while the halfer position implies that one should assign higher probabilities to experiencing an event before than after the loop.

Causal loops also raise questions about causation and free will. How can an event cause itself without violating the principle of causality? How can one act freely if one’s actions are predetermined by a causal loop? How can one avoid paradoxes or inconsistencies when traveling in time?

The application of the paradox to artificial intelligence and rational agents

Artificial intelligence (AI) is a branch of computer science that aims to create machines or systems that can perform tasks that normally require human intelligence, such as reasoning, learning, planning, decision making, and problem solving. AI can also raise various philosophical and ethical issues, such as the nature and limits of intelligence, the possibility and consequences of superintelligence, the moral status and rights of artificial agents, and the alignment of AI goals with human values.

The Sleeping Beauty Problem can be seen as a test case for AI and rational agents, which are systems that can act rationally to achieve their goals or preferences. The problem can be used to evaluate how well an AI or rational agent can handle situations that involve uncertainty, self-locating beliefs, imperfect recall, and multiple copies. For example, suppose that an AI or rational agent is put in a Sleeping Beauty scenario, where it is awakened once or twice depending on a random event and asked to assign a probability to that event. How should the AI or rational agent respond?

The answer may depend on how the AI or rational agent is designed or programmed. Different AI or rational agent models may have different ways of representing and updating their beliefs, preferences, and actions. For example, some models may use Bayesian networks, which are graphical models that encode probabilistic relationships among variables. Other models may use causal models, which are graphical models that encode causal relationships among variables. Some models may use utility functions, which are mathematical functions that measure the expected value or satisfaction of an outcome. Other models may use decision theory, which is a branch of mathematics that studies optimal choices under uncertainty.

The Sleeping Beauty Problem can also be used to explore how an AI or rational agent should interact with other agents who have different beliefs or preferences. For example, suppose that there are two AIs or rational agents who are put in a Sleeping Beauty scenario together, but they have different views on whether they should adopt the halfer or thirder position. How should they communicate and cooperate with each other? How should they resolve their disagreements or conflicts?

The Sleeping Beauty Problem can also be used to investigate how an AI or rational agent should deal with ethical dilemmas or moral responsibilities. For example, suppose that an AI or rational agent is put in a Sleeping Beauty scenario where it has to make a choice that affects the welfare of other agents or beings. How should it weigh the costs and benefits of its actions? How should it respect the rights and interests of others? How should it balance its own goals with social norms and values?

The challenges and open problems posed by the paradox

The Sleeping Beauty Problem is not only a puzzle, but also a source of insight and inspiration for various fields of inquiry. The problem reveals the complexity and subtlety of reasoning under uncertainty, self-locating beliefs, imperfect recall, and multiple copies. The problem also invites us to reflect on our intuitions, principles, and methods of rationality, probability, and belief revision. The problem also challenges us to explore the connections and implications of the problem for other areas of philosophy and science, such as epistemology, metaphysics, logic, ethics, psychology, physics, biology, and computer science.

However, the Sleeping Beauty Problem also raises many questions and difficulties that remain unresolved or controversial. Some of these questions and difficulties are:

• What is the correct solution to the problem? Is there a definitive answer or a conclusive argument for one position or another? Or is the problem underdetermined or indeterminate by the available information?
• What are the assumptions and implications of each position? Are they consistent, coherent, and plausible? Do they have any undesirable or paradoxical consequences?
• How should we evaluate and compare different positions? What criteria or standards should we use? How should we weigh the pros and cons of each position?
• How should we deal with variations or generalizations of the problem? How robust or sensitive are the positions to changes in the details or parameters of the scenario? How do they extend or apply to other cases or situations?
• How should we model or formalize the problem? What tools or frameworks should we use? How expressive or adequate are they for capturing the relevant features and aspects of the problem?
• How should we test or experiment with the problem? What methods or techniques should we use? How reliable or valid are they for measuring or eliciting our beliefs or preferences?

The Sleeping Beauty Problem is an open problem that invites further investigation and discussion. It is a problem that stimulates our curiosity and creativity, as well as our critical thinking and analysis. It is a problem that challenges our understanding and intuition, as well as our rationality and consistency. It is a problem that enriches our knowledge and perspective, as well as our imagination and vision.

The future directions and prospects for resolving the paradox

The Sleeping Beauty Problem is not only an open problem, but also a fertile problem that generates new ideas and directions for research and inquiry. The problem motivates us to explore new concepts and theories, to develop new models and methods, to discover new connections and applications, and to propose new solutions and arguments. The problem also encourages us to collaborate and communicate with others who share our interest and curiosity in the problem, as well as with others who have different views and perspectives on the problem.

Some of the possible future directions and prospects for resolving the paradox are:

• Developing a unified framework or theory that can accommodate both the halfer and thirder positions, or that can explain why they are both valid or reasonable in different contexts or respects.
• Finding a novel position or approach that can avoid or overcome some of the difficulties or objections faced by the existing positions or approaches, or that can offer some new insights or advantages over them.
• Applying the problem to other domains or disciplines where it can shed light on some existing problems or phenomena, or where it can inspire some new questions or hypotheses.
• Conducting empirical studies or experiments that can test some of the predictions or implications of the different positions or approaches, or that can measure some of the factors or variables that influence our beliefs or preferences in the problem.

So what?

The Sleeping Beauty Problem is a fascinating and intriguing puzzle that has captivated many minds and sparked many debates. The problem challenges our common sense and intuition, as well as our rationality and consistency. The problem also reveals the complexity and subtlety of reasoning under uncertainty, self-locating beliefs, imperfect recall, and multiple copies. The problem also invites us to explore the connections and implications of the problem for other areas of philosophy and science, such as epistemology, metaphysics, logic, ethics, psychology, physics, biology, and computer science.

The Sleeping Beauty Problem is not only a puzzle, but also a source of insight and inspiration. The problem motivates us to explore new concepts and theories, to develop new models and methods, to discover new connections and applications, and to propose new solutions and arguments. The problem also encourages us to collaborate and communicate with others who share our interest and curiosity in the problem, as well as with others who have different views and perspectives on the problem.

The Sleeping Beauty Problem is not only a source of insight and inspiration, but also a fertile problem that generates new ideas and directions for research and inquiry. The problem poses many questions and difficulties that remain unresolved or controversial. The problem also offers many possibilities and prospects for resolving or advancing the paradox. The problem stimulates our curiosity and creativity, as well as our critical thinking and analysis. The problem enriches our knowledge and perspective, as well as our imagination and vision.

The Sleeping Beauty Problem is a problem that deserves our attention and appreciation. It is a problem that challenges us to think deeper and broader. It is a problem that invites us to learn more and discover more. It is a problem that inspires us to create more and contribute more. It is a problem that awakens us to the beauty and wonder of philosophy and science.