Quanta 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.
Pradeep Mutalik in Quanta:
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 columnintroducing 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?