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## Saturday, 19 August 2006

During a brief, semi-lull in today (the last day!)'s graduate orientation, Matt Bedke introduced a paradox that I had not previously been aware of. The quick and dirty version of Newcomb's paradox (as it is called) goes something like this:
You are given a choice between either taking the content of Box A or the contents of both Box A and Box B. The content of Box B is known to you to be \$1000, but you do not have the option of taking only the content of Box B. The content of Box A is determined prior to your arrival by a psychic: if the psychic foresees you choosing both boxes, he places nothing in Box B; if the psychic foresees you choosing just Box A, he places a million dollars inside. The degree of the paradox is somewhat contingent upon the degree of accuracy attributed to the psychic by the storyteller, but in this instance Matt reported a 99% statistical accuracy rate.

The question arises: which choice is the most reasonable? If one takes both boxes, there's a 99% probability that they've chosen \$1000 over \$1 million; if one takes just Box A, there's a 1% chance that they've choses nothing over \$1000. Intuitions vary from person to person, and for some people from time to time. As I see it, it speaks to whether one reasons according to Bayesian probabilities or according to non-Bayesian deduction. The Bayesian sees 99% as near enough to certainty that choosing both boxes would be foolhardy; the chance that you are the one time in a hundred that the psychic is wrong is relatively slim. The non-Bayesian (which I think includes myself) reasons more according to the old maxim that "a bird in hand is better than two in the bush." Allow me to explain the two in more formal reasoning.

The Bayesian reasoner looks at the accuracy of the psychic and considers the option accordingly. Since the psychic is 99% correct, choosing both boxes means that there's a 99% chance that he did not place the million dollars in the box; likewise, choosing just Box A means that there's a 99% chance that he did place the million dollars inside. Thus, if one takes both boxes, it's a near certainty that they've chosen \$1000 over \$1 million, wherease if one takes just Box A, it's a near certainty that they've chosen \$1 million over \$1000. The latter scenario is far more reasonable than the former, so it seems more reasonable to always take just Box A.

Ignoring the accuracy of prediction previously achieved by the psychic, one can look at the situation as a basic dilemma: either Box A contains a million dollars or it does not; if it does not, then if one takes both boxes, they will have \$1o00, and if they take just Box A they will have nothing; if it contains the million dollars, then if one takes both boxes, they will have \$1.001 million, and if they take just Box A, they will have \$1 million. Either way, whether the psychic has placed the million in Box A or not, one is always better off taking both boxes. It seems quite reasonable to always take both boxes.

I'm sure there are other arguments for and against both mindsets, but these strike me as the most straightforward. I haven't read Nozick's article ("Newcomb's Problem and Two principles of Choice," in Essays in Honor of Carl G. Hempel, ed. Nicholas Rescher, Synthese Library (Dordrecht, the Netherlands: D. Reidel), p 115.), but I would be curious to see just how he fleshes this out and which approach he endorses.