Wolfgang Schwarz

Sleeping Beauty, Dutch Books and Newcomb's Problem

A curious aspect of the Sleeping Beauty debate is the role of Dutch Books. At first sight, it looks as if Dutch Book considerations support thirding (see e.g. Hitchcock 2004). However, as Halpern 2006 shows, Beauty can also be Dutch Booked if she is a thirder. Some have argued that these arguments might fail because in Sleeping Beauty type cases, credences and betting odds can come apart (see e.g. Bradley and Leitgeb 2006). I disagree. Instead, I will argue that her vulnerability to Dutch Books doesn't show that Beauty is irrational -- at least not if she is a halfer.

Let's begin with something completely different, a little game.

Two boxes contain either nothing or $60 each, depending on the outcome of a fair coin toss which you didn't see. One of the boxes is given to you, the other one to somebody else. If you open your box and there is money inside, you can keep it; if you open the box and it is empty, you have to pay $40. Moreover, if the other person opens her box and it is empty, you'll have to pay $40 for that as well. Assuming that you don't care about the other person, would you open your box?

It seems that you should. Here is the payoff matrix, assuming that the boxes contain $60 iff the coin landed heads (H):

The numbers in brackets give the expected utility of the lottery to the left. For instance, if you both open your boxes, there's a 50% chance that you'll get $60, and a 50% chance that you'll pay $80; the expectation value of this lottery is $-10. As the table shows, opening the box is the dominant strategy: no matter what the other person does, your expected payoff is greater if you open your box.

Of course, if you both open the box, your expected payoff is $-10; whereas if you had both not opened the box, it would have been $0. The game is a Prisoner Dilemma.

Let's turn it into a Twin Dilemma. Suppose that the other person is a clone of yourself, so that you can be confident that they will open their box iff you open yours. Would you open your box?

Evidential Decision Theory (EDT) says that you should not open your box: opening it is strong evidence that your partner opens her box; hence if you open your box, you can be confident that you get the $-10 lottery, whereas if you don't open it, you can be confident that the outcome is $0.

Causal Decision Theory (CDT) says that you should open the box: no matter what your clone does, opening the box is always the better option. If you could somehow cause your clone to not open her box by not opening yours, then of course you should do that. But the setup doesn't provide you with any remote controlling abilities. At no point in the game are you in a position to make it the case that neither you nor your clone opens the box. You can only control what you do. This means that if your partner actually opens her box, your choice is between the $-10 lottery (opening your box) and the $-20 lottery (not opening); and if your partner doesn't open her box, your choice is between the $10 lottery (opening) and $0 (not opening). Either way, you should choose the former.

I am convinced that CDT is right here, despite the fact that EDTers on average perform better in this game than CDTers. In fact, we can easily change the game so that CDTers will always lose:

Before making your decision, you are offered a bet that pays $140 iff there is money in the box, whether or not you open it. The bet costs $65.

You should obviously buy this bet. What happens when a CDTer is now paired up with their clone is that they always end up losing $5.

EDTers might ridicule CDTers. Isn't the game effectively Dutch Booking the CDTers, or at least pairs of CDTers? Moreover, unlike in Newcomb's Problem, where CDTers are confronted with an empty box and EDTers with a box full of money, this game isn't biased against CDTers. Everyone faces exactly the same situation; yet CDTers lose $5, while EDTers on average gain $5.

-- Well, no. In fact, the situation is a Newcomb Problem. For CDTers have been paired up with CDTers, and EDTers with EDTers; that's how the EDTers got an advantage. If a CDTer was paired up with an EDTer, she would on average win $15. But since she's paired up with a CDTer, she couldn't possibly do better than losing $5. What would have happened if she, being paired with a CDTer, had not opened her box? She would have faced an expected loss of $15. This would not have been rational.

The fact that CDTers are vulnerable to financial exploitation in this type of setup therefore doesn't undermine their rationality. Dutch Book arguments are meant to reveal an inconsistency in the agent's evaluative attitudes. In this case, at least, the argument doesn't do that.

OK. Now back to Sleeping Beauty.

On Sunday evening, Sleeping Beauty is put to bed. She knows that if a certain fair coin lands heads, she will be woken up on Monday and the experiment will be over; if the coin lands tails, she will be put back to sleep afterwards and woken up again on Tuesday, with all her Monday memories erased.

What should Beauty's degree of belief in heads be when she wakes up on Monday morning? Halfers say 1/2, thirders say 1/3.

Here is (the beginning of) the Dutch Book argument for thirding:

On each awakening, we offer Beauty a bet for $40 that pays $100 if the coin lands heads, and nothing otherwise. If Beauty assigns credence 1/2 to the coin landing heads, she ought to buy this bet. So if the coin actually lands heads, she will gain $60. If it lands tails, she will accept the offer twice and lose $80. If the experiment is repeated over and over, Beauty will on average lose $10 per trial.

Here it is assumed that Beauty will bet in line with her credences. I agree that this should not be taken for granted. In general, if someone believes a proposition p to degree d, one can expect them to pay up to $d for a bet that pays $1 if p and $0 otherwise. This is because her expected utility for the bet will normally be $1 * P(p) + $0 * P(~p) = $d. But it won't be so if p is probabilistically dependent on the bet (as when you're asked to bet that you won't bet today), or if the bet affects the value of the outcome (as when you believe you'll go to hell for betting).

So let's calculate the expected payoff for taking the bet. There are two possible situations: either the coin landed heads or it landed tails. If it landed heads, the bet will pay $100-$40 = $60. What if the coin landed tails? It then matters whether we use EDT or CDT. In EDT, the expected payoff is given by the probability of various outcomes conditional on accepting the bet and the coin landing tails. Conditional on tails, Beauty will face this same situation twice, once on Monday and once on Tuesday. And she can assume that whatever she does now, she will do it on the other occasion as well. Hence conditional on the assumption that she takes the bet and the coin landed tails, she will expect a total loss of $80. The expected value of betting is therefore 1/2 * $60 + 1/2 * $-80 = $-10; the expected value of not betting is $0. If Beauty is an EDTer, she won't accept the bet.

But Beauty should not be an EDTer. From the perspective of CDT, $-10 is the expected desirability, or news value, of betting, and $0 the desirability of not betting. But rational decisions should maximize not expected desirability, but expected (causal) utility. To calculate the expected utilities, we have to look at all the ways in which Beauty's decision might (for all she knows) be causally related to the possible outcomes.

1. One possibility is that the coin landed heads. In this case it must be Monday, and betting will deliver $60, while not betting will deliver $0.
2. Another possibility is that the coin landed tails and it is Monday. In this case the end result depends on what Beauty will do on Tuesday. Assume she will accept the bet on Tuesday. Then the net result will be $-80 if she accepts the bet now, and $-40 otherwise.
3. Another possibility is that the coin landed tails, it is Monday, but Beauty will not bet on Tuesday. Then the end result will be $-40 if she bets and $0 otherwise.
4. Alternatively, the coin might have landed tails and it is already Tuesday. Then the end result depends on what Beauty did on Monday. Assuming she accepted the bet then, the end result will be $-80 if she bets now and $-40 if not.
5. Finally, the coin might have landed tails, it is Tuesday, and Beauty did not bet on Monday. Then the outcome will be $-40 if she bets and $0 if not.

The expected utility of betting is the sum of the desirability of betting given these dependency hypotheses multiplied by the respective probability of the hypothesis.

The first hypothesis, that the coin landed heads, has probability 1/2. What about the other ones? Initially, Beauty might be undecided about whether she will bet or not. At this point, she will also be undecided about what she will do on Tuesday if it is now Monday and the coin landed tails (and about what she did on Monday if it is now Tuesday). The expected utility of betting will be 1/2 * $60 + 1/8 * $-80 + 1/8 * $-40 + 1/8 * $-80 + 1/8 * $-40 = $0; the expected utility of not betting will be 1/2 * $0 + 1/8 * $-40 + 1/8 * $0 + 1/8 * $-40 + 1/8 * $0 = $-10. She ought to accept the bet.

It is easy to check that no matter what Beauty believes about whether she will bet at the other day or not, she is always better off if she bets. In particular, on the assumption that she won't bet on the other occasion, her expected payoff for not betting will be $0, while her expected payoff for betting will be $10. And on the assumption that she will bet, her expected payoff for betting is $-10 and $-20 for not betting:

If Beauty is a CDTer, she will therefore accept the bet. Moreover, she will know that she will take the bet at the other occasion as well, if it arises. So her expected payoff will be $-10.

Again, we can turn Beauty's expected loss into a sure loss by selling her another bet that costs $65 and pays $140 iff the coin lands tails. (This is the second half of the Dutch Book argument.) If she accepts this bet, the outcome is

Does this show that Beauty is irrational? No. The situation is exactly parallel to the game above. If Beauty could somehow make it the case that she doesn't accept the bet at either day, she ought to do so. But she can only control what she does now. In this setup, the right thing to do is to accept the bet and pay the $5.

The upshot is that if Beauty is a halfer and a CDTer, then she is indeed vulnerable to financial exploitation. But that's not because she is irrational. It's because the Sleeping Beauty setup then turns into a Twin Dilemma, and thereby into a Newcomb Problem.