This was written in 2009 although I’ve known about this one-boxing solution for quite a while. Wittgenstein once said that you can tell that an itch is gone if you no longer find yourself returning to scratch it. This solution did it for me. My target is essentially the epigraph by David Lewis at the start of the essay. Lewis was a famous two-boxer and these words of his encapsulate everything that is wrong with two-box reasoning, in my opinion.
Thinking inside the boxes: Newcomb’s problem still flummoxes the great philosophers (2002), by Jim Holt. www.slate.com
Newcomb’s problem divides philosophers. Which side are you on? (2016), by Alex Bellos. www.theguardian.com
2. Why you should take just box B
Those who favour taking just box B are often called “one-boxers”. Well-known one-boxers include the philosophers Terence Horgan and Michael Dummett, not to mention William Newcomb himself.
Indeed, many people begin as one-boxers, even if they subsequently change their minds, because the case for one-boxing is so glaring, given how the problem is described.
Needless to say, the case rests simply on your belief (or expectation) that the predictor has predicted your choice correctly.
For, believing this, you should expect that if you take just box B, he will have predicted this and have left the million dollars in box B. Conversely, you should expect that if you take both boxes, he will have predicted this and have left box B empty; so you will just get the thousand dollars in box A.
In short, you should expect to get a million if you take just box B but only a thousand if you take both boxes and so you should be compelled to take just box B. This flows at once from your belief that the predictor has predicted your choice correctly.
There may be a concern with your degree of belief (or level of confidence) in the predictor’s ability since, as Newcomb’s problem is meant to be understood, you are not supposed to be completely sure that a correct prediction has been made but only “reasonably confident”.
This is as it should be, since complete certitude in the predictor’s powers is not a realistic state of mind. Indeed, someone might disarm the whole issue by denying that such certitude is possible! But if you are not completely sure that a correct prediction has been made, can the case for taking just box B still be made?
The answer is yes, because the case may still be argued through the traditional principle of maximizing expected utility.
Thus, suppose that the predictor is known to be 80% accurate, in the sense of having proved in the past to have predicted a person’s actions correctly 80% of the time. This is his track record, say, over a large number of trials involving people who have reacted to the problem in different ways.
This number will then represent your subjective confidence in a correct prediction. You should be 80% sure that he has predicted your choice correctly, as it were. More precisely, you should uphold these four conditional probabilities:
Prob (Box B contains a million | You take just box B) = 0.8
Prob (Box B contains nothing | You take just box B) = 0.2
Prob (Box B contains nothing | You take both boxes) = 0.8
Prob (Box B contains a million | You take both boxes) = 0.2
Assuming that the utility of money may be captured in dollars, the “expected utility” of taking just box B will then be:
(0.8 × $1,000,000) + (0.2 × $0) = $800,000
while that of taking both boxes will be merely:
(0.8 × $1,000) + (0.2 × $1,001,000) = $201,000
So a traditional expected-utility comparison recommends taking just box B.
Most one-boxers don’t need these numbers however. Provided one’s belief in the predictor is sufficiently firm, the qualitative argument is persuasive enough.
Indeed, given the terms of the problem, it is agreed on all sides that anyone who takes just box B may reasonably expect to find a million dollars therein, whereas anyone who takes both boxes may reasonably expect to go home with only a thousand. (The only disagreement comes from those who decline to have any faith in the predictor at all.)
Both sides agree on this, which makes one-boxers shake their heads in dismay at those who nevertheless think that it is better to take both boxes. How can anyone favour taking both boxes under these circumstances?