Question: Two Boxes
Answer: Luke Schroeder ’13
This is a tricky question, as all good trick questions are. The key to answering this question comes from keeping basic rules of causation in order. In this case, the experimenter’s prediction has no necessary causal power over the subject’s choice. It is true: the experimenter has an enormously high rate of success in predicting his subject’s choices. But from this it does not follow that the experimenter is always correct in his predictions—only that he’s very good at making them. Hence he could very well be wrong about what he predicts the subject will choose. This means that when we choose, there are four possible amounts of money we can receive. These amounts are based on the four possible combinations of selections and experimenter predictions. The possibilities are:
1) We pick both boxes, and the experimenter has predicted that we pick only B (which means that there is money in both A and B).
2) We pick box B, and the experimenter has predicted that we pick both boxes (which means that there is money only in A).
3) We pick both boxes, and the experimenter has predicted that we pick both boxes (which means that there is money only in A).
4) We pick box B, and the experimenter has predicted that we pick only B (which means that there is money in both A and B).
The results are as follows.
In possibility (1), we get the contents of both boxes—we get the thousand from A and the million from B. Remember that the experimenter is wrong, so our selection of both boxes doesn’t preclude the money from being in B. We made the best choice.
In possibility (2), we get only the contents of B—no money at all. Remember that because the experimenter is wrong, there is no money in B! Meanwhile, A had the thousand. We would have been better off picking both boxes. We made the worst choice.
In possibility (3), we get the contents of both boxes, and the thousand. The experimenter was correct, and so there never was any money in B. Picking B would have resulted in the experimenter being wrong, and us receiving no money—the situation described by possibility (2). We made the best choice.
And finally, in possibility (4), we get only the contents of B—the million. But meanwhile, A had the thousand as well. We could have picked both boxes, making the experimenter wrong, and received more money, as described in possibility (1). We made a poor choice.
In every one of these cases, we would have been better off picking both boxes. Such a choice does not lose, regardless of the experimenter’s prediction. It doesn’t matter at all what sort of prediction I think the experimenter is making. My success is relative—it depends only on how much money is actually in each box when it is time for me to choose. It has nothing to do with the experimenter being correct (unless what I should do is based on trying to prove him wrong, which could be an interesting way of having fun).
Hence I will select both boxes.
Perhaps the real dilemma here is this: suppose you are the experimenter. What will you predict?