A New Argument For Thirding

A solution to one of the world's most controversial paradoxes

The sleeping beauty problem is among the most controversial philosophical problems. When hearing about it, most people have extremely strong views about it, thinking one position is obviously correct. And yet there’s still widespread disagreement about which position is obviously correct. The puzzle was originally summarized by Adam Elga in the following way:

The Sleeping Beauty problem: Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?

There are two main answers: 1/2 and 1/3. Halfers say that you knew you’d wake up and you didn’t learn anything new, so you should remain at the default 1/2 probability of a coin coming up heads. Thirders say that because there are two wake-ups if tails, you should think that tails is twice as likely as heads. I’ve written lots of articles about it, including a paper defending the thirding view and an article about why the main halfing argument is wrong. But here I think I have a very simple and straightforward argument for thirding that is quite demonstrably correct (it’s broadly inspired by Cian Dorr’s argument, but simpler and, I think, more persuasive).

(This is the world’s greatest meme format).

Suppose that, like in the original case, a coin is flipped on Sunday. If it comes up tails, you wake up twice, once on Monday and once on Tuesday, each time you don’t have memories of the previous wake-up. If the coin comes up heads, you wake up once on Monday without memories, and a second time on Tuesday without memories. However, on Tuesday, after 15 minutes of waking up, the drugs will put you to sleep—suddenly and without notice—and you’ll sleep for the remainder of Tuesday.

After waking up, it seems obvious that you should be indifferent between heads and tails—both of them predict two wake-ups. Then, after 15 minutes, if you don’t fall asleep, you should have a credence of two-thirds that the coin came up tails—if it came up heads, there’s a 1/2 chance you’d be asleep by now, while if it came up tails, you’d definitely still be awake now.

I claim that this is relevantly like the original sleeping beauty problem. The only difference is that in the original, you’re asleep all day on Tuesday if the coin comes up heads, while in this version, you’re awake for 15 minutes. But surely this shouldn’t make a difference! To see this, suppose we make the waking time increasingly small. What if you’re only awake for half a second on Tuesday if the coin comes up heads? Then what should your credence be? Surely the half-a-second of waking shouldn’t make a difference to your credence! But this means that if you should third in this case where you’re awake for half-a-second on Tuesday if the coin comes up heads, you should third in the original sleeping beauty problem—if being awake for half-a-second is no different from being asleep all day in terms of reasoning, then necessarily your judgments can be the same.

So now the question is simply: should your credence in the coin having come up heads be 1/3 in the scenario described, where if the coin comes up heads, you’ll be awake for half-a-second on Tuesday. Seems like it should for two reasons. First, in the case where you’d be awake for 15 minutes on Tuesday if the coin came up heads before being put to sleep, it was intuitively obvious that if you’re awake for more than 15 minutes, you should think tails is twice as likely as heads. But this case is relevantly alike—the only difference is the amount of time one is awake for, but surely that shouldn’t make a difference!

Second, we can argue for a credence of 1/3 the same way we did in the case where you were awake for 15 minutes. In both cases, the odds you’d remain awake after the guaranteed wake-up period is 1/2 if the coin comes up heads, and 1 if the coin comes up tails. During the guaranteed wake-up period, you should think heads and tails are equally likely, so after you remain awake, you should think tails is twice as likely as heads.

Now, you might try to avoid the argument by suggesting that the wake-up only matters if the person thinks about anthropics during the wake-up. So if you wake up for one second, because you don’t have the time to think about anthropics, you never assign 1/2 credence to the coin coming up both heads and tails, so therefore you never do the Bayesian reasoning described in the previous section. Thus, the view would be that the wake-up only makes a difference if during the wake-up period you think about the anthropic situation.

But this is dicey for a bunch of reasons. First of all, the thing that matters isn’t actually if you performed the reasoning but what your credence should have been. Upon waking up, both heads and tails were equally likely, so now that you’ve gotten evidence (remaining awake) that’s twice as strongly predicted on heads, you should be at 2/3 on heads. Second, this gives bizarre results in the case where you’re awake for 15 minutes—it implies that if during the 15 minutes you didn’t think about anthropics, then after the 15 minutes pass, you should be at 1/2 on the coin coming up tails, rather than 2/3.

The only remaining option for the halfer is simply saying that being awake for half a second is relevant to how you should reason about it. On this picture, if you’re in the version of the scenario where on tails you’ll wake up on the second day for a single second, if you find yourself awake for a long time, you should think there’s only a 1/3 chance of heads. This view is quite hard to believe—whether you’ll wake up for a second on Tuesday or not shouldn’t make a difference to how you should probabilistically reason on the other days. If beauty stirred in her sleep for a second on Tuesday if the coin came up heads, that wouldn’t affect how she should probabilistically reason! To find out whether it makes sense to third, you don’t need to know whether Beauty will stir from her sleep and wake up for a second on Tuesday if the coin comes up heads.

We can give another argument against the relevance of the momentary wakeup that comes from Pruss. Suppose that the setup is as it was before, where if the coin comes up tails you’ll wake up twice with no memories, while if it comes up heads, you’ll wake up once with no memories and a second time with no memories for just a second. Suppose you wake up and remain awake for 10 minutes. You discover a button that will, if pressed, make it so that if it’s the first day and the coin came up heads, you won’t awake the second day. On this view, because the second-day momentary waking makes a difference to how you should probabilistically reason, even though you know that’s a state you’re not in, your credence should be different before and after pressing the button. This is super weird!

I think this argument is quite decisive. It joins the list of about a dozen completely conclusive arguments for thirding.