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?
In your case, one might make the case that what matters is whether the experimenter knows you. If she does, she will reason as you do and correctly predict that you will select both – leaving you with $1000 and nothing more. So why would it not be preferable to *tell* the experimenter (in all seriousness) that you plan to determine which option to take on the basis of some genuinely random process. Assuming you persuade the experimenter you are correct, she will also determine her choice randomly. That gives you a clear objective probability of taking both sums of money. If you use a (probability x reward) analysis (try it!) I think the expected outcome will be better than your proposal. Thoughts?
I see no grounds for the subject to ever attempt a random process. I think there are three basic reasons why a random process is sub-optimal.
First, the subject stands to lose in the worst fashion. Suppose his random choice indicates that he will select Box B, and that the experimenter’s random process leads to the prediction that he will select both. In this event, he receives no money.
Still, the subject’s *probable* earnings are high. The chances of winning no money are low. But these probable earnings mean nothing, because it is not true that the experimenter will determine her choice randomly. This is the second reason that a random process is suboptimal: that the experimenter will always make a conscious choice. I see no good explanation for why she should do otherwise. The evidence that she will always make a conscious prediction is considerable.
For one, she has been correct about her predictions 999 consecutive times. If making a random prediction was optimal, we could expect that she has made a random prediction on all 999 previous occasions. But it is highly unlikely that all 999 predictions were made correctly with random processes. If all 999 predictions were random, we would expect to see something on the order of 500 correct and 500 incorrect predictions. But we do not see this. Of course, it is logically possible that there could have been 999 random, correct predictions. But such an incident seems impossible, both to our basic understanding of random probabilities, and because the experimenter has reason to make conscious predictions.
More specifically, the experimenter can rationally expect that the subject will select both boxes. She knows that the subject is aware that this selection wins in each circumstance. She also has nothing to lose by predicting this selection. It is true, she might be incorrect. But she has never been incorrect and never will be. Why? She always holds a bigger threat value than the subject. The subject loses money if he selects box B (and the experimenter predicts otherwise) or if he drops out of the contest for any reason. This is a cost that is higher than he should be willing to bear. He has no reason to enter into or remain in this contest (nor does the experimenter have much reason to acquire or retain him for it) if he will receive no money. He always faces the opportunity cost of the $1000, and will consequently always make a selection sure to get him *some* amount of money.
This outcome seems unfortunate in its bounty, but for the third reason why a random process is suboptimal. The subject’s selection of box B (whether random or not) always produces at least a ½ chance of a losing choice. If he uses a random process, either he selects both boxes, in which case he gains the advantage I describe in the post, or he selects B, leaving him with the ½ chance at any success. Hence it is in the subject’s interest to make a conscious selection. For suppose that he randomly selects box B, and the experimenter randomly predicts this event. In this case, he earns the $1 million. Yet had he only made the conscious choice to select both boxes, he would have acquired the million and the thousand.
This interest forecloses upon the last possible reason for selecting randomly. And as we can see, upon no occasion is it in the subject’s interest to select box B, or employ any process that leads to this selection.
If the subject has any reason to suppose that the experimenter will predict the selection of both boxes, his selection of box B is *always* a losing affair. And as we have seen, there is reason. That is, the experimenter suffers no costs to any prediction, and holds the threat value in the situation. The subject is at the mercy of the prediction. Because of this, the interests of the subject and the experimenter coincide to produce 999 correct predictions. This coincidence does not stop for any occasion. The subject should always select both boxes.