What Should Sleeping Beauty Say, Logically Speaking?

Imagine that Sleeping Beauty agrees to take part in an experiment. First, she’s going to be given a special memory-loss drug that will make her forget what day it is. Then she’ll go to sleep. While she’s asleep, the people running the experiment will flip a coin. If the coin comes up heads, they’ll wake her up on Monday and ask her a question. If it comes up tails, they’ll wake her up on Monday, ask her the same question, give her the drug, let her fall asleep again, and then wake her up on Tuesday, asking the same question as before.

Because of the drug, however, she won’t know what day it is when she wakes up. Furthermore, she won’t know whether this is the first or second time she’s woken up. Each day will be a brand new experience for her. But each time Sleeping Beauty wakes up during the experiment, she’s told about the coin flip and the rule that says heads will mean she will wake up on Monday, and tails will mean she will wake up Monday and Tuesday. 

Now here’s the problem: Each time Sleeping Beauty wakes up and is told the rules of the experiment, she’s asked this question: “What are the odds that the coin flip came up heads?” That is, what are the odds that the coin came up heads and this is Monday (as opposed to tails and this is either Monday or Tuesday)? What should Sleeping Beauty say?

I came across this problem at the Quanta Magazine site yesterday. You may be surprised to learn that experts have been arguing about its solution for years:

The famous Sleeping Beauty problem has polarized communities of mathematicians — probability theorists, decision theorists and philosophers — for over 15 years…. 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… 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?

So, should Sleeping Beauty say there is a 1/2 chance that the coin came up heads, since there’s always a 1/2 chance that coin flips come up heads and a 1/2 chance that they come up tails? Should Sleeping Beauty be a halfer?

Or should she be a thirder and say there is only a 1/3 chance it was heads? After all, if it’s Monday, she was awakened because the coin came up heads or tails. If it’s Tuesday, she’s awake because the coin came up tails. That means it’s twice as likely she’s awake because the coin came up tails. There’s a 2/3 chance the coin came up tails and a 1/3 chance it came up heads. It sounds like Sleeping Beauty should be a thirder.

But not so fast! There was only that one coin flip on Sunday and we all know that coins have a 1/2 chance of coming up heads! Maybe she should be a halfer?

The Quanta article is fairly long and delves into why halfers and thirders give the answers they do, as well as why they often resist changing their minds. The comments that follow the article are even longer and include mathematical formulas. I didn’t read all the comments, and I’ve never studied probability, but I left my own comment anyway:

I wasn’t a halfer or a thirder before reading the article. Now I’m a committed thirder.

If you were to simply ask Sleeping Beauty whether a fair coin toss came up heads, she should say the odds were 1/2. Without any other information, that’s the rational answer. But you’re asking what odds Sleeping Beauty should assign, given the additional information she’s been given about the experiment. Since today could be either Monday or Tuesday (as far as she knows), it’s more likely that the coin came up tails. The fully-informed, rational answer she should give is 1/3.

So I think Sleeping Beauty should say this each morning: “Given how coin flips work, the odds are 1/2 that heads came up. But given how coin flips work and given what you’ve told me about this peculiar experiment, the odds are only 1/3 that heads came up. I can easily flip back and forth between the halfer and thirder positions, but why should I ignore the additional information you’ve given me? Taking into account what I  know about the experiment, I must conclude that the odds are 1/3 that heads came up. If that makes me a thirder, so be it. Now where’s my prince?”

But she shouldn’t take my word for it. She should make up her own mind.

The author of the article, Pradeep Mutalik, responded to my comment and everyone else’s last night. I’ll post his response tomorrow in case you haven’t already rushed to the Quanta site to read it.