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Newcomb's Paradox



The following is a slightly modified version of a post to the now defunct discussion group ASP (Association for Systematic Philosophy) — a group without which Wikipedia would probably not exist.


Newcomb's paradox, named after its creator, physicist William Newcomb, is one of the most widely debated paradoxes of recent times. It was first made popular by Harvard philosopher Robert Nozick. The following is based on Martin Gardner's and Robert Nozick's Scientific American papers on the subject, both of which can be found in Gardner's book Knotted Doughnuts. The paradox goes like this:

A highly superior being from another part of the galaxy presents you with two boxes, one open and one closed. In the open box there is a thousand-dollar bill. In the closed box there is either one million dollars or there is nothing. You are to choose between taking both boxes or taking the closed box only. But there's a catch.

The being claims that he is able to predict what any human being will decide to do. If he predicted you would take only the closed box, then he placed a million dollars in it. But if he predicted you would take both boxes, he left the closed box empty. Furthermore, he has run this experiment with 999 people before, and has been right every time.

What do you do?

On the one hand, the evidence is fairly obvious that if you choose to take only the closed box you will get one million dollars, whereas if you take both boxes you get only a measly thousand. You'd be stupid to take both boxes.

On the other hand, at the time you make your decision, the closed box already is empty or else contains a million dollars. Either way, if you take both boxes you get a thousand dollars more than if you take the closed box only.

As Nozick points out, there are two accepted principles of decision theory in conflict here. The expected-utility principle (based on the probability of each outcome) argues that you should take the closed box only. The dominance principle, however, says that if one strategy is always better, no matter what the circumstances, then you should pick it. And no matter what the closed box contains, you are $1000 richer if you take both boxes than if you take the closed one only.

One can make the argument for taking both boxes even more vivid by changing the setup a bit. For instance, suppose that the closed box is open on the face opposing you, so that you can't see its contents but an experiment moderator can. The moderator is watching you decide between one box and both boxes, and the money is there in front of his eyes. Wouldn't he think you are a fool for not taking both boxes?

Some proposed solutions are attempts to show that the situation as presented cannot occur. For instance, some say that it is impossible to predict human behavior with this kind of accuracy. That may very well be true, but even if it is physically impossible, that is not a satisfactory solution to a logical problem. Provided it is logically possible, we are still faced with a paradox.

Martin Gardner seems to be of the opinion that the situation is logically impossible. Basically he argues that, since the paradox presents us with two equally valid but inconsistent solutions, the situation can never occur. And he implies that the reason it is a logical impossibility has to do with paradoxes that can arise when predictions causally interact with the predicted event. For instance, if the being tells you that he has predicted you will have eggs for breakfast, why couldn't you decide to have cereal instead? And if you do have cereal, then did the being really predict correctly? He may very well have predicted correctly in the sense that, had he not told you about it, he would have been correct. But by giving you the information, he added something to the equation that was not there when he made his prediction, thereby nullifying it. *

So, just as there can be no barbers who shave only those who do not shave themselves, there can be no predictions that causally interact with the predicted events. Gardner is certainly right about this. What I don't think he is right about is in applying it to Newcomb's paradox. The being isn't telling you whether you will choose both boxes or not, so no self-defeating interaction is involved.

One might argue that there is a causal interaction at another level. The being is predicting that you will make a decision, so what happens if you refuse to do anything at all? This problem can, however, be circumvented very easily. The being can predict what you will do after he's told you of the experiment, and not have any contact with you (direct or indirect) between the time of his prediction and the time of the experiment. If you decide not to participate, he can predict you will do just that (and what he puts in the closed box is then irrelevant).

Although I disagree with Gardner, the notion of a prediction causally interacting with the predicted event plays a significant role in the difference between my view and Nozick's. Let's turn now to Nozick's analysis.

One possible solution that Nozick considers is the following: The dominance principle is not valid in Newcomb's paradox because the states ("1 million is placed in the box" and "nothing is placed in the box") are not probabilistically independent of the actions ("take both" and "take only closed box"). The dominance principle is acceptable only if the states are probabilistically independent of the actions.

Nozick disregards this solution by means of a counter-example: Suppose there is a hypochondriac who knows that a certain gene he may have inherited will probably cause him to die early and will also make him more likely to be an academic than an athlete. He is trying to decide whether to go to graduate school or to become a professional basketball player. Would it be reasonable for him to decide to become a basketball player because he fears that, if his decision were to go to graduate school, that means he probably has the gene and will therefore die? We certainly would not think that is reasonable. Whether he has the gene or not is already determined. His decision to become a basketball player will not alter that fact. And yet, here too, the probabilities of the states ("has the gene" and "does not have the gene") are not probabilistically independent of the actions ("decides to go to graduate school" and "decides to become a basketball player").

Both Gardner and Nozick conclude (though for different reasons) that, if they were faced with the situation presented by the paradox, they would take both boxes.

My solution to the paradox:

A paradox occurs when there are apparently conclusive reasons supporting inconsistent propositions. In the case of Newcomb's paradox, we have two arguments (both of which seem equally strong) for making opposite choices. The question is whether the paradox succeeds in making the opposing arguments equally strong. If it doesn't, then there actually is no paradox (or, to put it another way, the paradox will have been resolved). I don't think the two arguments are in fact equally strong, for, given the setup of the paradox, one can find a reasonable explanation for the alien being's ability to predict correctly. And in that case, the argument for taking the closed box is stronger than that for taking both boxes.

In order to explain why taking only the closed box is the more reasonable decision, let's first consider what it means to predict something. Prediction can mean at least one of two things. There's scientific prediction, where someone has observed similar conditions many times and predicts the outcome of a situation based on this experience (and on the assumption of some principle of uniformity in nature). This is how you predict that if you let go of your pencil it will fall, and how the weatherman predicts (though usually less successfully) what tomorrow will be like. And then there's prescience, where someone supposedly "senses" the future. Nostradamus isn't supposed to have been just a really good weatherman, he is supposed to have foreseen the future. This second kind of prediction is the equivalent of information traveling back in time. Now, whether the prediction is scientific or prescient, the solution to the paradox is essentially the same. But since I don't accept prescience, and because I don't think that that is how the paradox is usually understood, I will limit my explanation to scientific prediction only.

If the being predicts in the manner of a scientist, that means that there is a certain state of affairs, A, which holds at some point in time prior to your decision and the prediction, and which causes both. This connection between the prediction and the decision is what prevents your actions from being probabilistically independent of the states of the box. And it is realizing this that makes it rational to take the closed box only (i.e., it is what invalidates the dominance principle).

But now what about the hypochondriac? Why isn't his decision to play basketball just as acceptable? Here is where the interaction between the prediction and the event predicted comes in. In the case of Newcomb's paradox, there is no causal interaction (as already explained above). In the case of the hypochondriac there is. If the hypochondriac chose to go to graduate school without any knowledge of the gene's influence on behavior, then we would be justified in saying that his decision was in fact evidence for the presence of the gene. But because he knows about the gene's effects, he is in exactly the same situation as the person who is told by the being what he will have for breakfast. He now has an additional datum, not in the original prediction based on the gene, which in effect nullifies the prediction. Thus his choice to become a basketball player does not make him any less likely to have the gene.

Nozick and Gardner's choice to take both boxes, on the other hand, make them much less likely to make a million.


Notes

* See Does Determinism Imply Absolute Predictability? for more on this subject. ^


©1996, 2000 Franz Kiekeben
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