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Tag Archives: Newcomb’s Paradox

Philosophy has a long history of paradoxes, going back to the beginnings of philosophy in ancient Greece. Paradoxes are often concerned with a certain theory or system: Time travel paradoxes and physics would be a well known example. I like paradoxes. They push you to consider a problem from multiple directions. But what I like the most about them is how they require the listener to understand, and take apart, the language of the problem.

A rather popular paradox, sometimes described as a problem of free will and determinism, is known as Newcomb’s Paradox. I first encountered this paradox in Martin Gardner’s book of math puzzles, “The Collossal Book of Mathematics”. Newcomb’s paradox, named after it’s creator physicist William A. Newcomb, consists of a game between two agents. It is concerned with a branch of mathematics/philosophy known as decision theory. The two actors are the “Predictor” and the “Gambler”(My terminology). The situation is set up as such. There are two boxes, box B1 and box B2. Box B1 always contains $1,000, but B2 can contain either nothing or $1,000,000.

The Gambler, you in this situation, can either

1: Take what is in both boxes.

2: Take only what in in B2.

A certain amount of time beforehand, the Predictor guesses whether the Gambler will choose option 1 or option 2. If he guesses that the Gambler will choose option 1, he will leave B2 empty. On the other hand, if he guesses that the Gambler will choose option 2, he will put $1,000,000 in B2. The Predictor is so good at guessing that he is almost certain to be correct. In some cases the Predictor is described as being almost godlike in his accuracy. It isn’t necessary to assume any sort of determinism by the Predictor.

The paradox comes about by the fact that two, mutually exclusive, strategies appear to be correct. Assuming that the Predictor will almost certainly be accurate in his predictions, it would pay to always choose option 2. If you choose option 2, you will almost always earn one million dollars. A sure bet, if you will. In contrast, if you choose option 1, you will almost always earn only one thousand dollars and, only very very rarely, earn 1.1 million dollars. But consider, for a moment, that the Predictor made it’s guess a while ago. The cash is already in the boxes. So wouldn’t it actually be more logical to select option 1? If you assume that there is no “backwards causality”, that what you do doesn’t change, after the fact, the contents of the boxes then either the boxes contain the money or it doesn’t. So no matter what you actually choose, you can’t affect the odds. As such, it is better to select option 1, understanding that you are maximizing the cash you will earn by taking both boxes. This is because whatever IS already within the boxes will be there no matter what you choose.

Hopefully I’ve done a good job making both of the strategies sound compelling. Now when I first read the paradox, I started by trying to understand the key figure in this paradox, the Predictor. The paradox revolves around, in my mind, how the Predictor makes his predictions. The truth is that we aren’t told how he determines his choice. Thus we, the Gambler, are unable to get an idea of what his choice might be. If we assume no knowledge, we can only base our choices upon what we decide to do. Whatever we choose will most likely be what the Predictor will have chosen. This sounds weird but since the Predictor is good at guessing what we will do, we must assume that he will guess whatever we actually do because there doesn’t appear to be any better strategy. In this case the best strategy is to choose option 2. If, however, we understand him to be basing his choice, say, upon our psychological preference for one option or the other, we can see another potential strategy. Assuming the above information by the Predictor, we could attempt a strategy where we choose the opposite of our instincts. Either way, we can now see the paradox would seem to be the result of the definition of the Predictor.

There is obviously more to discuss about this paradox. What strikes me most strongly about this puzzle is that it seems to be a interesting point about determinism. If we assume that the Predictor does know everything you do in advance then he is essentially affecting the future by playing the game. We have defined the Predictor as making whatever choice you choose before you choose it. Yet, the earlier point that we cannot make money appear or disappear by our actions remains. To quote Martin Gardner’s excellent response on this topic.

“It is not logically inconsistent to suppose that the future is totally determined…but as soon as we permit a superbeing to make predictions that interact with the event being predicted, we encounter contradictions that render the existence of such a superpredictor impossible.”