**Experiment 1:**

A fair coin is flipped on Sunday, without you seeing the result, and then you go to sleep.

Tails: You get woken up Monday and Tuesday and each time shown a red flag. Your memory is erased each time, and you don't know whether it's Monday or Tuesday when you wake up.

Heads: You get woken up Monday and shown a red flag.

So, you're awake and see a red flag. What probability should you assign to heads? The two most common options are 1/2 and 1/3.

Experiment 1 is equivalent to the Sleeping Beauty problem, with a red flag added, which changes nothing.

**Experiment 2:**

Same as Experiment 1, except that on heads, you get woken up on both Monday and Tuesday, but on Tuesday you see a white flag.

It seems clear to me that in Experiment 2 you should assign 1/3 to heads. It's a standard Bayesian thing—you start with 1/2, and update on the additional information of a red flag: given heads, the chance of a red flag is 1/2, and given tails, the chance of a red flag is 1, so the red flag is evidence for tails, and the numbers work out to a probability 1/3 for heads.

I now claim that you should give the same answer for both experiments. This implies you should assign 1/3 to heads on Experiment 1.

Two arguments for equivalence. First, we can imagine a continuum of cases between Experiments 1 and 2, where the amount of awareness you have on Tuesday given heads varies continuously from zero (Experiment 2) to full human awareness (Experiment 1). Where exactly you would be on this spectrum on a white flag Tuesday does not seem to me to affect what credence is rational when you see a red flag and are fully humanly conscious.

Second argument. You can imagine that you're not told ahead of time whether you're in Experiment 1 or 2, but when you wake up and see a red flag you can press a button that you know has the following effect. If the coin was tails, the button has no effect. If the coin was heads, in which case it's Monday (since on Tuesday you don't see a red flag, if you're awake at all), and you press the button, then you'll wake up on Tuesday and see a white flag; if you don't press the button, you won't wake up on Tuesday in that case.

If you press the button, you're effectively in Experiment 2. If you don't, you're effectively in Experiment 1. If different credences of tails are appropriate in the two experiments, then how you decide about the button press *after* the coin toss affects what credence you should assign to tails. That's weird. (Is there a better choice of button press, one that will provide me with a better credence?)

Variant on second argument: You either do or do not see an independent random process depress the button when you wake up—you don't control the button. Should the outcome of this random process affect your credence about the initial coin toss? Certainly not: the outcome of this process is *ex hypothesi* independent of everything else. So, you should assign the same credence to tails in Experiments 1 and 2, and this credence should be tails.

I've been told that the claim that 1/3 is the right answer for Experiment 2 would be controversial. If so, then the argument only shows the credence is the same in the two cases, not that it's 1/3. But I think 1/3 is the right answer for Experiment 2.

## 4 comments:

Hi Alexander,

I think the problem with this puzzle is the ambiguity over what the sleeper is trying to do. Here are two versions of the problem using money:

Version 1 - A fair coin is flipped on Sunday, without you seeing the result, and then you go to sleep.

Tails: You get woken up Monday and Tuesday and each time you are given one pound. Your memory is erased each time, and you don't know whether it's Monday or Tuesday when you wake up.

Heads: You get woken up Monday and you are given one pound.

Version 2 - A fair coin is flipped on Sunday, without you seeing the result, and then you go to sleep.

Tails: You get woken up Monday and Tuesday and each time you are asked if the coin came up heads or tails; if you get it right you get a pound, if not, you get nothing. Your memory is erased each time, and you don't know whether it's Monday or Tuesday when you wake up.

Heads: You get woken up Monday and you are asked if the coin came up heads or tails; if you get it right you get a pound, if not, you get nothing.

In version 1 you get the money whatever you say. The probability you should assign to the coin being tails is the same as the probability you should assign to you winning one pound instead of two. This is clearly 1/2.

In version 2 you only get the money if you utter a truth. In this case it is clearly preferable to say it was tails because that will get you more money, but it won't change the probability.

In the original problem, when you are woken up you should assign a probability of 1/2 to both heads and tails. However, if you wish tomake more true utterances that you should always say tails.

Here's an argument that 1/3 is the credence in my Experiment 2. In Sleeping Beauty cases, the only two plausible credences are 1/2 and 1/3. Now, let's suppose you are in Experiment 2. You wake up. Before you open your eyes, however, you think about your credence. It's clearly 1/2. You open your eyes and see a red flag. Clearly this is evidence--you might have seen a white flag instead. It would be absurd to say that your credence should remain the same upon seeing the red flag.

Here's a more rigorous argument. If you were to see a white flag after opening your eyes, that would change your credence in heads to 1. But:

Theorem: If P(H|E)=1 and 0 < P(E) and 0 < P(H), then P(H|~E)< P(H).

So, not seeing a white flag after opening the eyes must decrease your credence (since 0 < P(white flag) and 0 < P(heads)).

Hence, after seeing the white flag, one's credence has to be less than 1/2.

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