My Talk About the Sleeping Beauty Paradox

A recent presentation

While everyone else was watching the election, my mind was on more important matters: the philosophy of probability. 

Yesterday, I gave a talk to my philosophy class about the Sleeping Beauty paradox. If you’re not familiar, this is the setup: 

On Sunday night, a fair coin is flipped. If the coin comes up heads, Sleeping Beauty will be woken up in her room once on Monday. If the coin comes up tails, she’ll be woken up on Monday, put back to sleep, given a memory-loss drug, and then woken up again on Tuesday. Each of the awakenings will be subjectively indistinguishable for her, and the memory-loss drug is perfect. The question is: When Sleeping Beauty wakes up, to what degree should she believe that the coin came up heads? 

The ‘halfers’ say: The coin is fair, so her ‘prior’ belief before she wakes up is that the probability of heads is a half. By waking up, Sleeping Beauty gains no new information; she always knew that she would wake up, and she can’t tell a heads-awakening and a tails-awakening apart. So, she should still think there’s a 50/50 chance the coin came up heads.  

The ‘thirders’ say: In all the instances of possible awakenings, in only one third of them will it be the case that the coin came up heads. Sleeping Beauty has no reason to privilege the hypothesis that she’s in any one of these cases rather than another; she should have equal belief in the possibilities {the coin came up heads and it’s Monday}, {the coin came up tails and it’s Monday}, and {the coin came up tails and it’s Tuesday}. So, she should think there’s a one in three chance the coin came up heads.

People are about equally split between these two possibilities, and everyone thinks that anyone who holds the opposite view is crazy (as with the Newcomb problem). The Sleeping Beauty problem has lots of implications across decision theory, the philosophy of probability, and even cosmology – which is why it’s a good subject for a talk. 

You can view my slides here. They should be followable without watching the talk and even (mostly) if you’re not familiar with the subject matter. The slides also contain a reading list. Similar to my talk about lead poisoning, I have no expertise here, and the presentation was informal and unrecorded. Several people told me in-person and online that they found the lead slides helpful, which is why I’m posting about Sleeping Beauty here. 

If you would like me to come and talk to your students/society/friends about any of my random side interests, you can email sam [at] thefitzwilliam [dot] com.

I’ll admit, it was a slightly uncanny experience to give a philosophy talk while the election news rolled in. Readers may be pleased to know I gave the example of a mispriced election forecast to explain the Dutch Book argument, and its relevance to the debate about whether the credences of rational agents must obey the Kolmogorov axioms. To my knowledge, neither candidate has yet expressed a view on this matter. 

[Update: I gave an updated version of this talk at the Asian Spring Program on Rationality.]