The Problem With The Sleeping Beauty Problem

School News Today
14 min readSep 11, 2023
The Sleeping Beauty Problem with Red Solo Cup

What is the problem with “The Sleeping Beauty Problem”? This video explains the history of the classic experiment, addresses what is wrong with it, and proposes a new solution.

The History of the Sleeping Beauty Problem

The Sleeping Beauty Problem is a thought experiment in decision theory that originated in ideas from economic game theory and a long-unpublished work by philosopher Arnold Zuboff of University College, London in the 1980s.

Philosopher Robert Stalnaker is often credited for being the first man to give the Sleeping Beauty Problem its name, but in his 2007 John Locke lecture at Oxford, he distanced himself from the scenario, commenting, “People are always performing these diabolical experiments on women. I don’t know.”

So we don’t know where this specific diabolic experiment originates.

But we do know that the Sleeping Beauty Problem as such was first made popular by Jamie Dreier, a philosophy professor at Brown University, in a puzzle chat group in 1999.

The Sleeping Beauty Scenario

Here is the premise from the original post by Jamie Dreier on the Usenet newsgroup rec.puzzles in March of 1999 and his explanation of its origin from an unnamed MIT graduate student:

We plan to put Beauty to sleep by chemical means, and then we’ll flip a

(fair) coin. 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.

The interview is to consist of the one question: what is your
credence now for the proposition that our coin landed Heads?

When awakened (and during the interview) Beauty will not be able to tell
which day it is, nor will she remember whether she has been awakened
before.

She knows the above details of our experiment.

What credence should she state in answer to our question?

-Jamie

p.s. Don’t worry, we will awaken Beauty afterward and she’ll suffer no ill
effects.

p.p.s. This puzzle/problem is, as far as I know, due to a graduate student
at MIT. Unfortunately I don’t know his name (I do know it’s a man). The
problem apparently arose out of some consideration of the Case of the
Absentminded Driver.

Adam Elga

A year after the Usenet group discussion, the problem became an academic sensation with the publication of Princeton professor Adam Elga’s paper “Self-Locating Belief and the Sleeping Beauty Problem.

Since that 2000 blockbuster, mathematicians, economists, and philosophers have written hundreds of articles proposing different solutions.

The titles of these papers and articles often feature wordplay on the fairy tale like “Laying Sleeping Beauty to Rest”, “Why the ‘Sleeping Beauty Problem’ Is Keeping Mathematicians Awake” or “Sleeping Beauty Should Remain Pure.”

Elga’s paper outlined two ways of calculating the likelihood that the outcome of the toss was Heads, and over time these two solutions have developed into camps.

Halfers

There are “Halfers,” led by American philosopher David Lewis of Princeton University, who posited that since any fair coin toss is equally likely to come up Heads or Tails, the chance the toss came up Heads is half regardless of other aspects of the scenario.

Thirders

And there are the “Thirders” who say that, like in the famous Monty Hall Problem, we must consider the context of the three coin tosses.

Because there are three possible awakenings, and only one happens if it comes up heads, the chance of the coin coming up heads is one in three.

Many alternatives have been added to the premise over time in an attempt to highlight different aspects of the problem.

Dualists

In 2008 Berry Groisman, mathematician and theoretical physicist at Cambridge University, introduced the Dualist perspective in a paper entitled, “The End of Sleeping Beauty’s Nightmares.” Groisman argues both positions are justifiable within the system of thought each employs.

Extreme Sleeping Beauty

In the Extreme Sleeping Beauty problem posited by Oxford philosopher Nick Bostrom, Sleeping Beauty is woken up a million times if the first toss comes up Tails.

In this scenario, it seems ridiculous to think the chances are half when she awakens at the end of the experiment that the coin came up Heads when we know there are a million more wakeups for Tails.

Monte Carlo

A Monte Carlo simulation illuminates this perspective.

If Sleeping Beauty is woken up a million times if the first toss comes up tails and she bets a dollar on each coin toss, she’d make much more money by always guessing tails than by always guessing heads.

Extreme Sleeping Beauty mirrors the 2003 argument Bostrom used to make the case that we are likely to be living in a simulation.

The Simulation Argument

The reasoning behind what is called “The Simulation Argument” is that very soon humans will have the capability to create a simulation of our universe, and when that happens it will be easy to create infinite copies of that simulation.

With the possibility of infinite copies, it becomes ever more likely we live in a simulation than in a base reality.

Time Warp

Yale research scientist Pradeep Mutalik even posits that Sleeping Beauty herself is in a time warp and conceives of the paradox as a Necker Cube, a picture that can be seen in two totally different ways at the same time.

New approaches to the paradox almost always involve pulling back the frame of the problem, expanding its context, or interrogating the questions within the problem itself.

Question the Premise

At an Institute for Computational and Experimental Research in Mathematics lecture at Brown University, Peter Winkler explains that the key to philosophical takes on The Sleeping Beauty Problem is questioning the premise.

“These philosophers, they question everything. I mean philosophers write about this problem and they question the concept of Tuesday. Everything is questioned.”

In that spirit, let’s question the story of Sleeping Beauty and ask: What Tuesday are we talking about?

Although much of the philosophical and mathematical speculation about the Sleeping Beauty Problem focuses on identifying temporal location, the problem itself is seen as ahistorical. The timeline is a decontextualized Sunday, Monday, Tuesday.

The Temporal Axis of Human History

But placed on the temporal axis of human history, the Sleeping Beauty story dates back to at least fourteenth-century France

And groundbreaking Comparative Bayesian phylogenetic analyses by Sara Graça da Silva and Jamshid J. Tehrani suggest ancient roots in India long before that.

ARN–Thompson–Uther Index

The tale is classified as ATU 410 in the ARN–Thompson–Uther Index, which is a catalog of folktale types used in folklore studies.

(See, “The Phylogeny of Little Red Riding Hood” by Jamshid J. Tehrani for the example chart in the video).

Penceforest

An early version of the tale, Penceforest, was an anonymous French fourteenth-century chivalric romance that told a fictional origin story of Great Britain and the Arthurian world.

In this Medieval version of the Sleeping Beauty Tale, a young prince, Troylus, finds the naked Sleeping Beauty, Zellandine, locked in a tower and agonizes about whether he can have sex with her without her consent.

This early printed transcription of the tale is very much about consent and rape.

Trolus wants to kiss Sleeping Beauty but Reason and Discretion interfere and say:

“Sir Knight, it is not proper for a man to enter a place where a maid is alone in privacy, without previous permission; and he knows he must not touch her while she is sleeping.”

Despite this convincing plea, Desire steps in and directs Troylus that his kiss will act like a medicine and revive Sleeping Beauty.

Under the guise of science then, Troylus kisses Sleeping Beauty “so many times that the infinite number has not been recorded.” Troylus then rapes Zellandine, taking her virginity and impregnating her.

The Sleeping Corpse Motif

As Rachel Fennell argues in The Transformation of the Sleeping Corpse Motif in Medieval and Early Modern Literature, “Rape is reframed as a medicinal quest.”

When Zellandine does not speak or wake up after the scientific assault, Troylus becomes frightened and worries that Zellendine could later accuse him of disloyalty.

But the so-called medicine of rape does not work. Zellandine does not recover, and she gives birth to a child while still unconscious.

Sleeping Beauty is finally woken by her own baby. When she comes to, Zellandine is confused and upset about what happened to her.

She is overwhelmed by horror at the loss of her maidenhood and the fact of the rape. Even when she later reunites with Troylus, Sleeping Beauty continues to grieve the loss of her virginity and the assault.

Pentamerone

An Italian version appears in Pentamerone, a collection of Neapolitan stories collected by poet Giambattista Basile and published in 1634. Basile served as a courtier and eventually became a “Count” himself.

In the Italian variation, “Sun, Moon, and Talia,” the Sleeping Beauty character falls into a deep sleep when a piece of flax gets lodged under her fingernail.

Her father places the sleeping Talia on a couch in a remote estate. But a King finds the beautiful sleeping Talia and rapes her.

Talia becomes pregnant with twins as a result of the rape and again gives birth while still asleep.

Metaethics of Necrophilia

Philosopher and Boston College Professor Mary Daly commented on the necrophilic nature of these rape stories in 1978 and drew attention to the half-state of the fairy tale heroine in her metaethics.

The fact that Sleeping Beauty is not really alive overlaps with the profitable early modern publishing industry question of whether women had souls or were truly human. Sleeping Beauty was always a Necker cube: human but not, alive but dead.

Romanticism

In the Romantic period, the idea of a sleep/death state was a central theme, reaching its apotheosis in Mary Shelley’s exploration of the scientific nature of being, Frankenstein.

But most literary Romanticism was fueled not by science but by folklore and myth in which life, death, and sleep have magical connections. The most famous example of Romantic appropriation of folklore is the Grimm Brothers.

The Grimm Brothers

The Grimm brothers became interested in researching historic narratives when they attended the University of Marburg.

They were not experts, but their interest was scholarly. Grimms’ 1812 Fairy Tale version of Sleeping Beauty was not meant for children.

In the first version the Grimm brothers record, collected from women in their town, the Sleeping Beauty character, Little Brier-Rose, wakes up on schedule from a 100-year sleeping curse just as an eligible unmarried prince appears and kisses her. The Prince does not ask for her consent in this story, either.

We will return to the Grimm Brothers later, but let’s keep following this tale through history.

Disney

Most illustrations of the current-day Sleeping Beauty Problem reference pink Disney-style princesses. So let’s look at the Disney era and its place on the axis of historical time.

Women’s Rights in 1959

Most important, the 1959 Disney movie version of Sleeping Beauty retains the 15-year-old girl character and makes her into the underage heroine of the piece.

Disney’s Sleeping Beauty is not old enough to give consent any more than the Thought Problem Sleeping Beauty is. Fifteen is below the age of consent.

Let’s look at 15 in 1959.

In 1959 if the Prince had sex with her, Sleeping Beauty was likely to get pregnant. In 1959 it was illegal for unmarried women, let alone girls, to use birth control. It was not until 1972 that unmarried women could legally use the birth control pill.

If a Disney-era Sleeping Beauty wanted to catch up on the education she missed while sleeping for one hundred years, the odds were not good that she would get a university degree.

In the 1950s women’s enrollment in colleges had still not recovered from the preferential admission for men under the GI Bill or Servicemen’s Readjustment Act. It was not until Title IX in 1972 that schools receiving federal money were prevented from discriminating against women.

And in 1959 if Sleeping Beauty wanted to enter a doctoral program to research the probability of her being awakened twice, the odds were bad. Only 5% of STEM Phds were earned by women in the 1950s.

If she were in the tiny cohort of female Phds, Sleeping Beauty would have had trouble financing her education. On marrying the Prince in 1959, Sleeping Beauty would not have had legal access to her own credit, a right women only earned in 1974 with the Equal Credit Opportunity Act.

In 1959, Sleeping Beauty would need to ask her husband the Prince’s permission to open a bank account or get a credit card.

As for rape, it was not until 1993 that marital rape was considered a crime in all 50 states. So when they were married, the Prince would have had a legal right to have sex with Sleeping Beauty without consent.

As a traveling soldier, the Prince may well have had an STD. If he passed it on to Sleeping Beauty in the 1950s, under the government’s American Plan she could have been legally detained in a reformatory, workhouse or jail. If you don’t know about this dark chapter in American history that continued through the 1970s, ask yourself why.

Now let’s bring the story to the present day on the historical time axis.

Women in the University in The Current Day

The blockbuster Princeton philosophical paradox asks us to imagine that we are Sleeping Beauty.

Let’s put ourselves in the position of a woman in a university setting in the current day.

According to a study by the American Psychological Association,

More than 1 in 13 students reported being drugged, with women more than twice as likely to be victims of the crime. Sexual assault is the most common motive and result of being roofied.

According to the National Intimate Partner and Sexual Violence Survey:

More than 15 million women reported an alcohol or drug-facilitated sexual assault in her lifetime.

According to Bureau of Justice Statistics Research on campus violence:

The probability that a woman in college will be sexually assaulted is one in five.

A Center for Disease Control study estimated that:

About 2.9 million women in the U.S. experienced a rape-related pregnancy during their lifetime.

A report published in Nature by the US National Academies of Sciences, Engineering, and Medicine, found “Sexual harassment is pervasive throughout academic science in the United States, driving talented researchers out of the field and harming others’ careers.”

Philosophy is no exception.

(See our video How To Not Sexually Harass Your Students).

Self-Location in the Modern Academy

I’m not sure that a woman in a hypothetically better future would want to be located in this time at all.

But the question of the scientific Sleeping Beauty Problem is: how do we self-locate at all? How can we ever know what time we are in?

Much discussion of the Sleeping Beauty Paradox revolves around the question of a “Centered World.” According to David Kellog Lewis that means we have a possible world, an agent in it, and a time in that world.

Hierarchical Individualism

The story of Western philosophy is based on hierarchical individualism which assumes a white male observation point. But the Sleeping Beauty paradox encapsulates another type of dissemination of knowledge operating without a center.

In the Paradox, the possible world of Sleeping Beauty is complicated by the fact that it exists inside multiple other possible worlds in the form of variations on the tale in literary time.

Which world is really the center of any tale?

What is the Center of The Tale? An Experiment

Let’s imagine a thought experiment for a group of male philosophers and mathematicians.

You are to perform a chemical experiment on an unaccompanied underage girl that involves putting her to sleep.

After you perform the experiment, the underage girl will put you to sleep and give you an injection that makes you forget everything that happened.

Don’t worry, you won’t suffer any ill effects.

The girl will then explain the experiment to you and ask you what your credence is that the year is:

In Medieval times

In Victorian times

The Golden Age of Hollywood or

The current year

To avoid any temporal tips in your environment, in the experiment, you will all be wearing the timeless graduation robes and doctoral hoods of your institution and the experiment will take place in a Medieval-style room at Oxford University.

The 15-year-old will be naked as she was in the Medieval Sleeping Beauty story but covered in a plain linen sheet for her protection.

The Drunkards’ Walk

Humans are notoriously bad at being able to judge probability. Theoretical physicist and mathematician Leonard Mlodinow of CalTech explains this in The Drunkards’ Walk: How Randomness Rules Our Lives.

The primate brain is not wired for the task of predicting odds.

Mlodinow shows that even highly trained professionals such as doctors are surprisingly bad at guessing probabilities and understanding statistics.

So despite your high level of expertise, we are going to give you some questions to direct your investigation into the odds of Sleeping Beauty awakening you in the present day versus Medieval times or the 1950s.

Questions for the Experiment

Are women’s minds still considered a blank slate for men’s projections?

Are men still in a position of knowing (what day it is, what the result of a coin toss is)?

Are women still in a position of presumed ignorance?

Do men think for women? Do they reason for the Sleeping Beauties?

Are women treated as equals in science?

Is the classroom an equal place for women?

Is women’s humanity a given?

These rhetorical questions point to a different way of looking at the Sleeping Beauty Problem in history.

Is it possible that the Sleeping Beauty experiment in its various permutations is part of a simulation as Nick Bostrom hinted?

Could the real subject of the experiment be the scientists themselves and not Sleeping Beauty at all?

Could this be a simulation designed to teach future experimenters about the historical nature of patriarchy in the 21st century?

Is the answer to this question knowable or is it a reflection of Fitch’s Knowability Paradox? Is it an “unknowable unknown truth”?

Are We Living In a Simulation From the Future?

How could we possibly calculate the odds of being in a simulation for the scientist who wakes up in this experiment?

The odds are high that there are multiple simulations at work.

First, there are the many calculations on the question of the likelihood that we are all the subjects of a simulation, performed in the future, with odds ranging from 1% to 99% but most often calculated at 50%.

Simulations from the Past?

But what about simulations from the past?

There has always been a historic simulator hidden inside the Sleeping Beauty Problem.

Language as Simulation

First, there is the problem of language itself which is a kind of animal simulation machine.

Philosopher Michel Foucault is usually credited with the idea that language shapes reality.

But since Foucault was an advocate of sex with minors, and is accused of raping pre-pubescent Tunisian boys in graveyards, we will not credit him.

Instead, let credit the Sapir-Whorf hypothesis of linguistic relativism, which predates Foucault. Studies by Lera Boroditsky have validated the idea that language shapes perception with practical demonstrations.

The setup of the experiment already begins within the shaping of language.

We are used to factoring by X to ignore the universal element of human language in experiments, but linguistic simulators are always at work.

Fairy Tales as Simulation

Fairy tales are complex linguistic simulation engines constructed from the building blocks of mammalian verbal modeling of the world.

Included in the simulator that is ATU 410 is the problem of sexual reproduction. The presence of babies in these tales forces us to pull back our frame again and look at the first simulation mechanism, which is sex.

Sexual Reproduction as Simulation

When Sleeping Beauty awakes in the folk tales, she has reproduced and has now simulated herself and her rapist in the form of one baby or matching twins.

Zuboff’s Cell-by-Cell Reproduction

In Zuboff’s original philosophical musing on the paradox, he meditates on the question of an imagined cell-by-cell reproduction of self and what it means for self-location within a many-worlds interpretation of quantum mechanics.

Zuboff believes, “There is no exclusive now centering on one time.”

When Zuboff’s article was finally published in 1990, the idea of an exchange of cells between people was pure science fiction.

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