[Math Lair] Newcomb's Paradox

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Newcomb's Paradox is a paradox formulated by William Newcomb that was first brought to the attention of the philosophical community in 1969 by philosopher Robert Nozick. Consider the following scenario: There are two closed boxes, B1 and B2, on a table. B1 contains $1,000. B2 contains either nothing or $1 million (you do not know which). You may choose from one of the following two options:

  1. Take the contents of both boxes
  2. Take the contents of box B2 only.

At some prior time an entity who is able to make highly accurate predictions about your decisions and/or human behaviour in general (and whom you know is able to do so) has made a prediction regarding what you will decide. If the entity predicted that you would take both boxes, he has left box B2 empty. If he predicted that you would take only B2, he has put $1 million in it. So, should you choose (a) or (b)? Assume that, if the entity believes that you will avoid making a choice in some manner, such as by flipping a coin, he will leave B2 empty.

The paradox lies in the fact that there are valid reasons for choosing either (a) or (b). If you take both boxes, the entity will almost certainly have anticipated that and left B2 empty, whereas if you take only B2, the entity will almost certainly have anticipated that, and put $1 million in B2; therefore, it would be best to take only B2. However, either the money is already in B2 or it isn't. The contents of B2 will not change once you make your choice. So, whether there is $1 million in B2 or not, you will always make $1,000 more by choosing both boxes.

This is a well-known philosophical paradox, often seen in discussions about free will, but there is a mathematical side to it as well. In game theory, we might say that there is a conflict between the "expected-utility principle" and the "dominance principle". Assuming that the being is able to predict with near certainty, the expected utility of taking only box B2 is much larger (you're very likely to take $1 million, whereas if you take both boxes, you're very likely to only get $1000). On the other hand, no matter what is in the boxes, the option to take both boxes dominates the option to only take box B2.

Sources used (see bibliography page for titles corresponding to numbers): 5.