Newcomb's Paradox / Problem

One of the most simply stated but astonishing of the so-called prediction paradoxes that bear on the problem of free will. It was devised in 1960 by William Newcomb, a theoretical physicist at the Lawrence Livermore Laboratory and the great-grandson of the brother of the astronomer Simon Newcomb, while contemplating the prisoner's dilemma.

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 superior being, with super-predictive powers that have never been known to fail, he has run this experiment with 999 people before, and has been right every time. has put $1,000 in box A and either nothing or $1 million in box B. The being presents you with a choice:

(1) open box B only, or
(2) open both box A and B.

The being has put money in box B only if it predicted you will choose option (1). The being put nothing in box B if it predicted you will do anything other than choose option (1) (including choosing option (2), flipping a coin, etc.).

The question is, what should you do to maximize your winnings? You might argue that since your choice now can't alter the contents of the boxes you may as well open them both and take whatever's there.

This seems reasonable until you bear in mind that the being has never been known to predict wrongly. In other words, in some peculiar way, your mental state is highly correlated with contents of the box: your choice is linked to the probability that there is money in box B.

These arguments and many others have been put forward in favor of either choice. The fact is there is no known "right" answer, despite the concerted attentions of many philosophers and mathematicians over several decades.

The Solution:

The solution to exposing Newcomb's paradox as fallacy is to view the potential monetary gains against the probability of the beings prediction being correct. For a given probability P the best choice is the one that gives the greatest return.

The formula:

Return = P(Correct) + (1-P)(Wrong)

Lets assume a probability of 1 (ie. the being has a 100% chance of predicting correctly).

Return from taking B:

1(1,000,000) + 0(0) = 1,000,000

Return from taking both A and B:

1(1,000) + 0(1,001,000) = 1,000

As can be seen, if the being has a 100% chance of predicting correctly then the rewards of taking box B are much more favourable than taking both box A and box B. You may notice that this outcome is equivalent to Argument 2 above. Now let's assume a probability of 0.5 (the being has a 50% chance of predicting correctly).

Return from B:

0.5(1,000,000) + 0(0) = 500,000

Return from A and B:

0.5(1,000) + 0.5(1,001,000) = 501,000

With a probability of 0.5 the rewards of taking both box A and box B are slightly greater than taking only box B. This means that if the being only has a 50% chance of correctly predicting your choice then you should take both box A and Box B. Now let's assume a probability of 0 (The being has no chance of predicting correctly).

Return from taking B:

0 + 1(0) = 0

Return from taking both A and B:

0 + 1(1,001,000) = 1,001,000

This time the rewards of taking both box A and box B are far greater than taking only box B.

You may have noticed that the two formulas Return = P(1,000,000) + (1-P)(0) and Return = P(1,000) + (1-P)(1,001,000) are in fact straight lines and can be graphed as such:

The two lines intersect when P is at a value of 0.5005. That is to say that if the being has a greater probability than 0.5005 of being correct then you should take box B. If the probability of the being being correct is less than 0.5005 then you should take both box A and box B.

The paradox should now be exposed to you as a fallacy. If it is not, let me state more clearly why. Argument 1 relies on an implicit assumption that there are equal chances* whereas Argument 2 asserts that there is every chance that the being will correctly predict your choice.

At the end of the day you could say the being is either going to predict correctly or not, ascribe a probability of 0.5 and take both box A and box B. But if the being had previously made 1000 correct predictions then surely this line of action would be 1000 times more foolish than the other? If you say no then I guess you'd give yourself a 50/50 chance of beating Garry Kasparov at chess?

* Argument 1 asserts that no matter what you choose, 'the being either has or has not put the money in box B'. It follows from this assertion that if you choose box B then there may or may not be money in box B. Likewise, if you choose box A and box B there still may or may not be money in box B. This is equivalent to saying that the being may or may not correctly predict your choice.