2008년 4월 21일 월요일

Newcomb's paradox

Newcomb's paradox is an interesting thought experiment. Some philosophers think it has to do with the problem of free will and causality. Some think it is silly. There are many slightly differing formulations, and some think the details of the formulation matter a lot. I don't take the paradox too seriously and I don't think the details matter, nevertheless it is very interesting to observe people's reaction when they hear this paradox for the first time.

So here is my version of the paradox.

You are presented with two boxes, A and B. The box A is transparent, and inside it is $1,000. The box B is opaque. You are presented with two choices. Either you can take both boxes, or you can take the box B only. And then there is the predictor. With high accuracy the predictor predicts your choice. If he/she/it predicts you will take both boxes, it puts nothing in the box B. If he/she/it predicts you will take the box B only, it puts $1,000,000 in the box B. What is your choice?

Argument 1. The box B has either $1,000,000 or nothing. The predictor already predicted my choice and decided the content of the box B, so my choice doesn't affect the content of the box B in any way. Taking both boxes is always superior to taking the box B only, by the amount of $1,000. Therefore I take both boxes.

Argument 2. Since the predictor is highly accurate, if I choose both boxes it is highly likely that I will find the box B to be empty, and that my income will be $1,000. Since the predictor is highly accurate, if I choose the box B only it is highly likely that I will find the box B to have $1,000,000, and that my income will be $1,000,000. Therefore I take the box B only.

Note that the first argument is independent of the predictor's accuracy. Also note that the mechanism of the predictor's prediction is completely unspecified, and you have no way to assess its accuracy other than that it is "highly accurate".

Let's change that. Suppose that you are an observer, and you observed 10 (100, 1000, whatever) people choosing either both boxes or one box. Suppose that you observed the predictor's prediction to be always correct. Would it change your choice? Does the number of observations matter? Note that if you support the first argument, you have no reason to change your choice.

Also note that unless the mechanism of the prediction is specified, you can argue that the past performance is no proof of the future performance, and to think otherwise succumbs to the gambler's fallacy. The gambler may think that if coin toss turned head ten times in a row, the next toss is likely to turn head (or tail). Of course if coin is fair, each toss is independent and head and tail are equally likely, no matter what.

Suppose that you observed the predictor's prediction to be 90% correct. It is easy to cacluate that the expected value for both boxes is $101,000 and for one box is $900,000. Would the fact that the predictor is not perfect change your choice? Does the exact probability matter? Does the ratio of the amount of money matter? If your choice is based on the expected value, they probably do.

Some argues that the predictor as presented in the paradox can not exist. Suppose that people in the paradox are replaced with AI programs, and the predictor has much faster computer to run programs. Then it is obvious to me that the predictor can indeed predict the behavior. Since I accept the Turing thesis (which states, in short, human-like AI is possible), I also accept that such predictor can exist.

Also note that the predictor in the paradox does not need to be perfect. Public opinion polls routinely make predictions such as that females are more likely to vote for the candidate X, that people who live in Y are more likely to vote for the candidate Z, etc. How can you argue that it is impossible to predict the human behavior?

(P.S. So what is my choice? I choose the box B. My reason shall remain private though.)

댓글 1개:

James :

Sanghyeon,

Thanks for taking the time to elaborate on the paradox you mentioned after class a couple weeks ago. Rereading it, it now certainly seems to make a lot more sense to me to choose A, for the reasons you mentioned (that my choice cannot retroactively affect what is in the box, and I cannot therefore lose money by taking both). In fact, because the predictor makes its prediction and takes (or does not take) action before I am presented with my choice, then it seems to me that it should be irrelevant to my choice.
I am more curious now why you would go with B, but perhaps I shall remain so.