Editor’s note: In January, we ran an Insights column about the much-debated Sleeping Beauty problem. Now, our puzzle columnist Pradeep Mutalik claims to have discovered why this problem is so polarizing. In the spirit of experimentation, we will be inviting a panel of experts to weigh in on whether this insight adds any new clarity to the problem. Think of it as an informal Polymath-style “peer review.” Readers are also invited to share their opinions in the comments section below.
The famous Sleeping Beauty problem has polarized communities of mathematicians — probability theorists, decision theorists and philosophers — for over 15 years. In the puzzle, the fairy-tale princess participates in an experiment that starts on Sunday. She is told that she will be put to sleep, and while she is asleep a fair coin toss will determine how the experiment is to proceed. If the coin comes up heads, she will be awakened on Monday, interviewed, and put back to sleep, but she won’t remember this awakening because of an amnesia inducing drug she is given. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday, again without remembering either awakening. In either case, the experiment ends when she is awakened on Wednesday without being interviewed.
Whenever Sleeping Beauty is awakened and interviewed, she won’t know which day it is or whether she has been awakened before. During each awakening, she is asked: “What is your degree of certainty that the coin landed heads?” (“Degree of certainty” is sometimes expressed as “belief,” “degree of belief,” “subjective certainty,” “subjective probability” or “credence.”) What should her answer be?
This simple mathematical problem has generated an unusually heated debate. The entrenched arguments between those who answer “one-half” (the camp called “halfers”) and those who say “one-third” (the “thirders”) put political debates to shame. In my column introducing the problem, I compared it to a Necker cube, the popular visual illusion that can be perceived in two completely different ways. But while most people can flip quite easily between the two views of the Necker cube, halfers and thirders tend to remain firmly rooted in their view of the Sleeping Beauty problem. Both camps can certainly do the math, so what makes them butt heads in vain? Is the problem underspecified? Is it ambiguous?
It’s hard to believe that this simply stated problem has remained open for over 15 years. It doesn’t seem to be underspecified — both camps feel confident that they have solved it. This suggests some deep ambiguity that causes smart people to disagree vehemently. In my solution column, I discussed some dichotomies in the two camps: Halfers count experiments, while thirders count awakenings; halfers calculate from the experimenter’s point of view, thirders from Sleeping Beauty’s. But these are mathematical techniques that both camps know how and when to use. When halfers and thirders reach different conclusions, it is clearly not a matter of mistakes in calculation: They must, in effect, be solving two entirely different problems.
This point — that the problem is ambiguous and both sides are correct — has been made by several people, including our own prizewinning reader Josh, who cited a paper by Berry Groisman. Groisman showed that there are two interpretations of the problem that are both consistent under standard probability theory. I agree, but this doesn’t explain why both halfers and thirders are so strongly convinced that only one side can be right. This phenomenon occurs, as Groisman suggests, because this quarrel is not about mathematics. Rather, it is a fight about two subconscious ways of understanding the problem’s statement, each linguistically valid and intuitively appealing. The fact that there are linguistic issues implying two different propositions here was pointed out by reader Nathan Gantz in his comment on our original Sleeping Beauty column. I show below that these two divergent approaches arise from different construals, or meanings, of the phrase “landed heads,” which refers to an event that happened in the past. These two meanings arise in human speech and writing whenever anything is expressed about a different time, and specifically whenever the past tense is used. In everyday communication we can figure out the correct meaning from context, but in the Sleeping Beauty problem both meanings are available for people to latch onto at an intuitive level.
To reveal these two meanings, let me add a small detail to the story:
Imagine that when the coin was tossed on Sunday, it was mounted on a brass plaque in the position it landed in, so that the result of the coin toss can be checked at any time. This plaque is kept in a locked safe in Sleeping Beauty’s room.
Certainly this act cannot make any difference to the logic of the actual problem. But it primes our intuitions, enabling us to see that the original question can be interpreted in two ways:
Meaning 1: The Action Interpretation
“What is your degree of belief that the coin landed heads?” = What is your degree of belief that the coin landed heads in the act of tossing? (Imagine the coin being tossed.)
Note that the belief, though current, is about a previous event: The verb “landed” is in the past tense. Whenever a past event is evoked in speech or writing, the listener or reader has to decide how much of the event’s background is relevant. Sometimes, a phrase referring to the past requires the listener to “import” the event’s background without the speaker’s explicitly saying so — a “frozen past tense.” It’s like a photo taken when you were 10 years old, showing your old house in the background, even though you’ve changed a lot since then and the house is gone. Here’s a question about the past that requires this kind of implicit background importing: What is your belief that my friend the rock star spent a full year’s pay on his first guitar? This question refers to your belief about the money my friend was making when he bought the guitar, not what he makes today. After all, he is a rock star now, 20 years later. In a similar way, the first meaning of the Sleeping Beauty proposition imports its background act, which can be intuitively accessed by invoking the image of the coin being tossed. It refers to the probability that the coin landed heads when it was tossed: obviously one-half.
Meaning 2: The Property Interpretation
“What is your degree of belief that the coin landed heads?” = What is your degree of belief that the preserved coin in the safe is showing heads now? (Imagine which side is showing on the coin in the safe.)
Here the past-tense predicate “landed heads” is used as a way to describe a property previously gained by the coin you are referring to today. You are no longer concerned with how likely it was to have landed heads or tails when originally tossed, but only with the likelihood that the coin being referred to now, the preserved coin, shows heads and therefore “landed heads” at some time in the past (as opposed to landing tails). The same past-tense verb construction is used, but now it does not import the background action.
As an example, consider that until a few months ago, six of the nine U.S. Supreme Court justices were graduates of Harvard Law School and the other three were graduates of Yale Law School. If I had asked you then, “What is your degree of belief that a random U.S. Supreme Court justice attended Yale Law School?” you would probably have answered one-third. Here, the question is obviously not intended to import the background of the event: I was not asking you about the odds that the justice chose Yale Law School out of all the law-school choices available at the beginning of his or her career, but rather, I was using “attended Yale” as a property of the person within the current group of justices, as opposed to the opposite property, “attended Harvard.” Similarly, the second meaning of the Sleeping Beauty proposition does not import the past action, but merely looks at heads as a present property, previously acquired. It can be intuitively accessed by invoking the image of the coin on the plaque in the safe, and refers to the probability that the coin is showing heads now, supposing you had to bet on it. This probability could be anywhere from 0 to 1. In this case, there are three possible cases where Sleeping Beauty could be examining the coin on the plaque: A Monday when the coin shows heads, a Monday when the coin shows tails, and a Tuesday when the coin shows tails. Therefore the probability of it showing heads is one-third.
For each real world question similar to the Sleeping Beauty question, context makes it clear what is expected: whether we should invoke the background of the event (the friend’s previous salary) or merely use the acquired property (the justice’s law school affiliation). However, the wording of the question in the original problem provides no context to force one interpretation or the other. Both are up for grabs — and boy are they grabbed tightly by the two different camps! To be fair, language understanding usually occurs automatically and subconsciously, so few of us question the ready-made interpretations that our minds settle on.
Halfers, I suggest, consciously or subconsciously find the first interpretation more salient, use it in their modeling, and come up with a value of one-half. Thirders subconsciously prefer the second interpretation, base their calculations upon that, and come up with a value of one-third.
How can this be? Isn’t the coin in the safe the same one that landed heads or tails on Sunday?
Let’s ask the sophisticated and intelligent princess Sleeping Beauty, who is well versed in the natural arts and sciences such as linguistics, math and science (not to mention fauna and flora). Let’s catch her at the time of her interview with the experimenter’s assistant at one of her awakenings.
Experimenter’s Assistant: What is your degree of belief for the proposition that the coin landed heads?
Sleeping Beauty: That question is ambiguous: It can be interpreted in two different ways. Do you mean my belief about the likelihood of heads in the act of tossing the coin on Sunday, or do you mean my belief about the likelihood of heads being shown on the preserved coin in the safe?
EA: But the coin in the safe is the same coin that was tossed on Sunday, and it shows the same result.
SB: Yes, but you can have a different belief about the probability of heads in the act of tossing a fair coin (which is always one-half at the time of tossing) and the probability of the same coin showing heads some time later.
Let’s suppose that while walking on the seashore, I see 15 coins, 10 of which show tails and five of which show heads. Perhaps a boy who was on the beach before me took away half of the coins that came up heads because he liked that side. No matter how many coins I gather, I always find two showing tails for every one showing heads. Half of the coins are, to me, lost in space. Now my level of belief that a new coin I encounter will show heads and therefore “landed heads” at some time in the past is only five out of 15, or one in three. I can only base my belief on the clear-cut and reliable statistics of the coins I encounter. Maybe one day, I’ll find the boy’s stash of coins that landed heads — and if I do, my expectation that there are equal numbers of coins that landed heads will change to one-half — or maybe I never will. Notice that we use the verbs “landed” or “came up” in two senses: for the act of tossing the coin, as in “the coin just landed heads,” and for the act of finding it later, as in “here’s a coin that shows heads and therefore landed heads sometime in the past.”
Here’s a different situation. Imagine I have a specific kind of double vision: The only objects it affects are coins showing tails. When a coin shows heads, I see it as one coin; when it shows tails, a strange optical effect makes me see double. I actually saw only 10 coins, five showing heads and five showing tails. But my strange affliction, which is unknown to me, causes me to see heads and tails in a ratio of 1-to-2. Hence I have an expectation of one-third that any new coin I find will show heads, and therefore landed heads. If I find out later that I have this condition, I can correct my erroneous, but previously valid, belief. Of course, knowing that these are all fair coins, I never waver in my belief that they originally landed heads one out of two times in the act of tossing.
Now imagine that I find the same 15 coins, but I’ve entered a time warp without knowing it, and five of the tails I saw were the same ones I had seen before. All the coins showing heads somehow escaped the time warp. Now half of the heads are lost in time, or you could say that the tails are doubled in time. Again, my belief that a new coin I encounter will be heads is, validly, one-third. If and when I come out of the time warp, and realize it, I will change my belief back to one-half.
I don’t want you to think that such distortions necessarily reflect some kind of error of judgment. It may be that half of all coins that landed heads self-destructed on landing so that the ratio of coins that I find reflects the existing ratio in the world. As long as I am unaware of any systematic errors, I therefore have to trust the ratio that I find as the basis for my belief in the probability of the coins I am likely to find.
Thus, distortions in time, space and perception that I am unaware of, or coins that exist for different lengths of time, or any systematic process that alters the frequencies of the two coin-toss results differentially, can alter the relative frequencies I reliably find. All these processes influence my valid belief regarding the proportion of the coins I am likely to encounter that landed heads.
Let’s return to your question, which is actually two separate questions. The first question is: What is my degree of belief that the coin landed heads in the act of being tossed? This value, of course, was one-half on Sunday, and will remain one-half until I actually find out what happened. I am a true halfer about this.
The second question — what is my degree of belief that the preserved coin is showing heads now — is the same as saying, “What is my degree of belief that the coin in the safe landed heads sometime in the past?” On Monday and Tuesday, it is one-third, because I am in a time warp with half the heads lost in time. I am definitely a thirder in response to this question when I am awakened on these two days.
When I emerge from the experiment’s time warp on Wednesday, the value of my belief in heads will return to one-half, because the two different interpretations will coincide. Then I expect that your boss, the professor, will tell me how the coin actually landed. At that time, my belief that the coin landed heads will settle on exactly 0 or exactly 1.
EA: Wow, all that sleeping must be good for the brain! Ouch, my head hurts. I think I’ll take some of the amnesia drug I’m about to give you…
That’s all there is to it, halfers and thirders. The rest, as they say, is just plain math.
After attaining the above perspective, I can honestly say that I can clearly see both views of the Sleeping Beauty Necker cube, and can readily flip from one view to the other. Now that you’ve heard the argument, here are three questions to answer:
- Can you now disentangle the two different propositions implicit in the Sleeping Beauty problem?
- If you were previously a committed halfer or thirder, can you now see the validity of the other point of view?
- Do you think that this discussion throws any new light on the problem?
As usual, the best comment will earn its author a Quanta Magazine T-shirt. If a comment offers an idea that enhances this argument, it may earn the commenter an invitation to be a co-author if this argument is submitted to a journal for publication.
 Note that, as a couple of language experts including Steven Pinker of Harvard University have told me, this is not a phenomenon specific to a particular language. How much background we need to assume for a given statement is a universal problem and is usually resolved by context.
 The numbers in this example have been rendered obsolete by the death of Justice Scalia. I have kept the original numbers because of the relevant ratio they generate.
Note: This column was updated on April 8, 2016, to mention a reader comment that brought up linguistic issues. The columnist elaborates on the issue of assigning credit in the comments below.