Newcomb’s Paradox

Unknown-1In the realm of mathematical puzzles and thought experiments one can find a stock pile of paradoxes.   The Mariam-Webster Dictionary defines a paradox as “an argument that apparently derives self-contradictory conclusions by valid deduction from acceptable premises.”  One short example of such oddities is the Liar’s Paradox.  It is the deceptively simple declaration, “This sentence is false.”   After some evaluation, one concludes that, if the sentence is false, it is also true.  Likewise, if it is true it must be false, providing a self-contradiction that does not permit the assignment of a single truth value.

Another mind-bending conundrum is Newcomb’s Paradox.    The paradox was first created by William Newcomb at the University of California Lawrence Livermore Laboratory and later first published as a philosophy paper by Robert Nozick in 1969.  In a review by Olle Häggström of William Eckhardt’s book Paradoxes in Probability Theory in the March 2013 edition of Notices (a publication of the American Mathematical Society), Häggström describes Newcomb’s Paradox as follows:

Newcomb’s Paradox

An incredibly intelligent donor, perhaps from outer space, has prepared two boxes for you: a big one and a small one. The small one (which might as well be transparent) contains $1,000. The big one contains either $1,000,000 or nothing. You have a choice between accepting both boxes or just the big box. It seems obvious that you should accept both boxes (because that gives you an extra $1,000 irrespective of the content of the big box), but here’s the catch: The donor has tried to predict whether you will pick one box or two boxes. If the prediction is that you pick just the big box, then it contains $1,000,000, whereas if the prediction is that you pick both boxes, then the big box is empty. The donor has exposed a large number of people before you to the same experiment and predicted correctly 90 percent of the time, regardless of whether subjects chose one box or two. What should you do?

The interesting thing about this paradox is that picking the bigger box alone does not guarantee a payoff; since there is still a 10 percent chance not in one’s favor based on past experiments that the intelligent donor is wrong.   In the same vein, there will always be the payoff of $1000 by picking the smaller box, either alone or with the bigger box.  However, in the case of picking both boxes, there is a 10 percent  chance in one’s favor based on past experiments  that the intelligent donor is wrong.   So, with this in mind, what would be your optimal choice?

About Avery Carr

Avery Carr is a senior analyst and past senior editor for the American Mathematical Society Grad Blog. He and his wife, Alison, live in Olive Branch, MS.
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