2 Newcomb's problem

Suppose you are in a room, with an opaque blue box and a transparent box in front of you. Both of the boxes contain a certain amount of money (which might be \$0). In the case of the transparent box, you can see that it holds \$1{,}000, but you can’t see what’s inside the blue box.

The choice you must make before leaving is whether to take the blue box alone (One Box), or whether to take the transparent box as well (Two Box). What should you do?

If that’s all there is to the story, and you prefer more money to less, the answer is obvious: you should Two Box. No matter how much money is in the blue box, Two Box will give you \$1{,}000 more.

Now imagine that we add some more detail to the case. Last night, while you slept, a demon scanned your brain and predicted what you would do in Two Boxes. If the demon predicted you would take one box, they put \$1{,}000{,}000 in the blue box. If the demon predicted you would take both boxes, they left the blue box empty. The demon has done this millions of times to other people in the past (including the brain scan, and including letting those people know about the brain scan before they made their choice between the boxes). The demon has always been right before: it’s very good at working out how people will behave based on looking at scans of their brains.

What should you do now?

Here’s one way of thinking about it. Whatever the demon may or may not have predicted last night, that’s now all in the past. Either the \$1{,}000{,}000 is in the blue box now, or it isn’t. There’s nothing you can do to change it now. So your choice is really between however much money is in the blue box, or however much money is in the blue box, plus \$1{,}000. Two Box makes you better off by \$1{,}000 no matter what. So — take the extra money!

Here’s another way of thinking about it. People who take one box almost always end up millionaires. People who take both boxes almost always end up with only one thousand dollars. This has happened millions of times before, and none of the people who thought they would be better off by doing Two Box actually ended up millionaires — they ended up with a comparatively pitiful \$1{,}000. A million is way better, and you’re not actually going to get the top prize of \$1{,}001{,}000 if you Two Box, whereas if you One Box you can expect to go home with a cool million dollars. What you care about is actually getting the money. So — take one box and enjoy being rich!

The way we’ve been doing decision theory up until now: setting up decision tables, choosing what seem to be the right states of nature, and applying things like Dominance principles, seems at first sight to support choosing Two Box. Taking the states of nature to be what the demon predicted last night, our decision table would look like this:

Taking both gets you \$1{,}000 more in either column. So taking both boxes strongly statewise dominates taking only the blue box.

However, even though these might intuitively seem to be the right states of nature—after all, it’s now out of your control whether the demon put one million dollars in the blue box last night or not—this can be disputed. Recall the conditions on states of nature from week 1:

• (i) Together with the act, they resolve the uncertainty and guarantee one outcome.
• (ii) The act you choose does not affect whether the state occurs, or how probable it is.
• (iii) They are mutually exclusive.
• (iv) They are jointly exhaustive.

Conditions (i), (iii) and (iv) are satisfied by the acts and states of nature in Table 1. Condition (ii), however, is not. For it really imposes two requirements: that your choices do not affect which state of nature eventuates and that choosing one act or another does not change how probable the states of nature are. It turns out that these two requirements are incompatible in cases like the Newcomb case we’re discussing now.

The first requirement is satisfied by the acts and states of nature set out in Table 1. The second requirement is not.

In fact the approximations are more than we need. All that matters for what follows is that \Cr(D \mid B) > \Cr(D \mid B^c): taking both boxes is evidence that the demon predicted you would.

This does not prove the states in Table 1 are wrong, or that we need to abandon dominance reasoning. Dominance reasoning of some sort seems to be secure. The question is: what requirements need to hold on a set of states of nature in order for dominance reasoning using those states to be legitimate? In other words, the dispute about whether to One Box or Two Box is also a dispute about what the requirements on states of nature should be. Condition (ii), as we incautiously stated it in section 2.6, can be read as requiring two things that are incompatible:

• (a) Probabilistic independence.
• (b) Causal independence.

This second notion isn’t as clearly defined as the first. But it is intuitive.

A good way to see why causal and probabilistic independence can come apart is to look at a causal graph (sometimes called a causal diagram). In a causal graph, the arrows represent the directions of causal influence: if there is an arrow from A to B, that means that A has a causal influence on B. Let’s see the causal graph for the Newcomb case, as we set it out:

As is indicated in the diagram, your brain state last night has a causal influence on your choice of whether to One Box or Two Box today, because what choice you make tracks neural and psychological features that can, in principle, be picked up by a very smart demon with a very advanced brain scanner. Your brain state last night also has a causal influence on the demon’s prediction, and in turn, on the contents of the blue box. That is: if you were in a brain state that meant you were likely to One Box the next day, the demon put the \$1{,}000{,}000 into the blue box; whereas, if you were in a brain state that meant you were likely to Two Box the next day, then the demon put nothing in the blue box.

Your choice today and the contents of the blue box are evidentially linked—choosing One Box now should raise your credence that the demon put \$1{,}000{,}000 into the blue box—because they share a common cause. It is your brain state last night that ultimately determines both; and your choice now gives you some evidence about your brain state, which is therefore also evidence about what’s in the blue box.

However, there is (at least, intuitively) no causal link between your choice today and the contents of the blue box. This is because your brain state last night has a causal influence on whether you One Box or Two Box today, but the reverse is not true: which choice you make today does not (at least, intuitively) have a causal influence on what your brain state was in the past. Past events can causally influence future events, but present or future events do not (at least, intuitively) causally influence past events. Evidential links are, in general, symmetric. Causal links are often asymmetric.

The question of what you should do in the Newcomb case splits decision theorists into two main camps. (There are others, but there are two main camps.) The evidential decision theorists think that you should One Box in the Newcomb case, and they hold that when applying dominance reasoning, the states of nature need not be causally independent but must be probabilistically independent. The causal decision theorists mostly think that you should Two Box in the Newcomb case, and they hold that when applying dominance reasoning, the states of nature need not be probabilistically independent but must be causally independent.

So, we have two different decision theories, based on two different, and conflicting, independence requirements on the set of states of nature: Causal Decision Theory (CDT) and Evidential Decision Theory (EDT).