## Tuesday, March 17, 2015

### Decision theory, evidence and Sleeping Beauty

I've been thinking about the following variant of Sleeping Beauty. A coin is tossed on Sunday, out of your sight. As usual, if the coin is heads, you'll wake up on Monday, and then sleep through until Wednesday. If it's tails, you'll wake up on Monday and Tuesday. But this isn't standard Sleeping Beauty. Your memory of the Monday wakeup won't be erased on Tuesday. Instead, you will be given a drug that makes it impossible for you to update your credence as to heads on Tuesday.

You wake up. It's Monday. You know it's Monday, because you don't remember an earlier wakeup. How should you set your credence?

Evidentially speaking, it's clear and uncontroversial. Your credence in heads evidentially should still be 1/2. A Monday wakeup is no evidence for or against heads. (Now, a Tuesday wakeup would a different matter—it is conclusive evidence against heads, but it would be evidence you are unable to update on due to the drug.)

But suppose that both on Monday and, if it's tails, on Tuesday you will be offered choices from a single broad and diversified portfolio of bets regarding whether the coin landed heads. Suppose, further, that you will be unable to decide except on the basis of maximizing expected utility (with respect to your credences). Then decision-theoretically, you should assign credence 1/3. A simple argument is that if the experiment is repeated, then in 1/3 of the times you're choosing from the portfolio it will in fact be heads and 2/3 of the times you're choosing from the portfolio it will in fact be tails (remember that once you set the credence on Monday, it won't be able to change on Tuesday). So you should gamble as if you had credence 1/3, and to do that you need your credence to be 1/3 since I assumed that you cannot but bet on the basis of your credence.

Interestingly, the same result follows if you're maximizing total lifetime expected epistemic utility with respect to a proper scoring rule: You should assign 1/3 to heads.

Yet evidentially your credence should be 1/2. This illustrates the fact that when you expect your future credences to have a chance of being irrational—because of your inability to update on Tuesday—then we have a conflict between what, on the one hand, the evidence supports and what, on the other hand, decision theory and epistemic utility maximization support.

The original Sleeping Beauty case, where you can't tell if it's Monday or Tuesday because your memory has been erased, has some similarity to this. For while in my modified case, Tuesday's credence is forced by a drug to be the same as Monday's, in the original Sleeping Beauty case Tuesday's credence is forced to be the same as Monday's due to memory loss and the fact that, presumably, you will make up your mind in the same way given the same data.

This similarity suggests that we should be suspicious of concluding that evidentially your credence should be 1/3 from the fact that both decision-theoretic and epistemic utility considerations lead to 1/3 in the original Sleeping Beauty case.

I only want to make this modest point. I think that's the only point the analogy supports. The analogy is not strong enough to support the conclusion that one should assign 1/2 in the original Sleeping Beauty case. But it is enough, I think, to show that cases like Sleeping Beauty are going to be exceptions to the correspondence between evidential and utility (whether pragmatic or epistemic) considerations.