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Is predicting human behavior inherently paradoxical? August 7, 2010

Posted by Ezra Resnick in Computer science, Game theory, Puzzles.
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Suppose that someone has a method for predicting how people will behave in certain controlled situations. We need not assume perfect clairvoyance; let’s say that past performance has shown the method to be 90% accurate. The predictor invites you into a room where you are presented with two closed boxes, marked A and B. The rules are as follows: you may either take both boxes, or take box B only. Box A definitely contains $1000; the contents of box B, however, depend on the prediction that was made (in advance) by the predictor. If he predicted that you would take both boxes, box B was left empty. If, however, he predicted that you would take only box B, $1 million were placed inside it.

This is called Newcomb’s paradox, created by the theoretical physicist William Newcomb in 1960. Why is it a paradox? At first glance, it seems obvious that you should take both boxes. Regardless of the contents of box B, taking both boxes always gives you an extra $1000! This is the principle of dominance in game theory, and it is difficult to dispute. However, an equally well-founded principle is that of expected utility: when you know the probabilities of the various outcomes, you should attempt to maximize your expected winnings. If you take both boxes, there is a 90% chance you will end up with $1000 (the predictor was right and so box B is empty), and a 10% chance of ending up with $1,001,000 (the predictor was wrong). So the expected utility when taking both boxes is $101,000. On the other hand, if you take box B only, there is a 90% chance of getting $1 million, and a 10% chance of getting nothing, so the expected utility is $900,000! It seems you should take only box B, then. But isn’t it irrational to leave behind the guaranteed $1000 in box A? And round and round we go…

As is often the case, the root of the paradox is self-reference. (Is the sentence “this sentence is false” true or false?) In Newcomb’s paradox, the subject knows his actions have been predicted, and this knowledge influences his decision. Therefore, the predictor must take his own prediction into account when making his prediction. In other words, if the predictor wishes to create a computer model which will simulate the subject’s decision-making process, the model must include a model of the computer itself — it must predict its own prediction, implying an infinite regress which cannot be achieved by any finite computer. This inherent limitation is reminiscent of Rice’s theorem in computability theory: it is impossible to write a computer program that can determine whether any given computer program has some nontrivial property (such as the well-known undecidable “halting problem”).

It is tempting to dismiss Newcomb’s paradox as logically impossible, under the assumption that we can never predict with any accuracy the behavior of a person who knows that his actions have been predicted. (Asimov’s Foundation introduces “psychohistory” as a branch of mathematics which can predict the large-scale actions of human conglomerates, but only under the assumption that “the human conglomerate be itself unaware of psychohistorical analysis.”) Nevertheless,¬†William Poundstone (in Labyrinths of Reason) describes a way of conducting Newcomb’s experiment that doesn’t require a computer model to model itself. It does, however, require a matter scanner. We would use the scanner to create an exact copy of the entire setting, atom by atom — two rooms, two sets of boxes, two subjects. (Neither subject knows whether he is the “real” subject or the clone.) We would then use the actual decision made by the first subject as our prediction for the behavior of the second subject.

So what would you choose? Box B, or both boxes?

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