Newcomb's Paradox

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Originator:

William Newcomb

Publication:

Nozick, Robert (1969) "Newcomb's Problem and Two principles of Choice", in Essays in Honour of Carl G. Hempl, ed. Nicholas Rescher

The Paradox:

There exists a being who has a perfect track record in predicting human behaviour. This being has two boxes (A and B) and offers you a choice of either taking box B or taking both box A and B. Box A is open and contains $1,000 while box B remains closed.

The being has predicted which choice you will make and has acted according to his prediction. If he has predicted that you will take box B then he has placed $1,000,000 in it. Otherwise, if he has predicted that you will take A and B then he has placed nothing in box B.

The paradox occurs because there are supposedly two valid arguments, one for taking box B and the other to take both A and B.

Argument 1:

The being either has or has not put the money in box B. If the being has put the money in B then taking B results in $1,000,000 whereas taking A and B results in $1,001,000. If the being has not put the money in B then taking B results in $0 whereas taking A and B results in $1000. In both cases the best option is to take both A and B and therefore you should take both A and B. This is shown in figure 1 below.

Figure 1

Argument 2:

The being has been correct in all his previous predictions and is therefore likely to predict correctly again. If the being predicts correctly then taking A and B results in $1000 whereas taking only B results in $1,000,000. Therefore you should take B. This is shown in figure 1 below.

Figure 2

Thus two apparently acceptable arguments lead to a paradox.

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.