25

Newcomb’s Paradox

25.1 SOME HISTORY

In 1950 the mathematicians Merill M. Flood (1908–1991) and Melvin Dresher (1911–1992), while working at the RAND Corporation in Santa Monica, California (an air force think tank), jointly created a puzzle question in game theory that has bedeviled analysts ever since. I’ll first give it to you in its best-known, nonprobabilistic form and then in the form that gives this last entry in the book its title (and in which probability makes an appearance). There is no solution section for this puzzle because, as I write, there is no known analysis that satisfies everybody. That’s why it’s the last problem in the book!

The best-known version of the Flood-Dresher puzzle is called the prisoner’s dilemma, a name given to it by Albert W. Tucker (1905–1995), a Princeton University mathematician. Imagine that you and another person have been arrested and each of you has been charged with two crimes, one serious and the other not so serious. You’ve both vigorously claimed innocence, but are now being held in separate cells awaiting trial. There is no communication possible between the two of you. Just before the trial is to start, the prosecuting attorney from the DA’s office shows up in your cell with the following offer.

There is sufficient circumstantial evidence to convict both of you of the not so serious charge, enough to get each of you a year in prison even if neither of you confesses. But if you confess, then the other person will be convicted of the more serious charge and get ten years in prison, and you will be set free. When you ask if the other person is getting the same offer, the answer is yes, and, further, when you ask what happens if both of you confess, the reply is that then both of you will get five years in prison. The puzzle question is now obvious: what should your decision be, to confess or not?

To help keep all the conditions clear in your mind, the following table of your various potential fates should help:

To make your decision, you might use the following standard game theory reasoning. The other person will either confess or not. It will be one or the other, and which one it is has nothing to do with anything you can control. So, suppose he or she does confess. If you confess you get five years, and if you don’t confess you get ten years. Clearly, you should confess if he or she confesses. But suppose your partner in crime doesn’t confess. If you confess you go free, and if you don’t confess you get one year. Clearly, you should confess if your partner doesn’t confess. That is, you should confess no matter what the other person decides to do. For you to confess is said to be (in game theory lingo) the dominant decision strategy.

But here’s the rub. The other person can obviously go through exactly the same reasoning process as you’ve just done, to conclude that his or her choice is also dictated by the dominant decision strategy of confessing. The end result is that you both confess and so you both get five years in prison! The paradox is that perfectly rational reasoning by each of you has resulted in a nonoptimal solution because if you both had simply kept quiet and said nothing, then you both would have gotten the much less severe sentence of one year in prison. Philosophers have argued (for decades) over whether this is really a paradox or merely “surprising,” and the literature on the problem had, even years ago, grown to a point where nobody could possibly read it all in less than that ten-year prison sentence. And it continues to grow ever more voluminous even as I write.

It was while thinking about the prisoner’s dilemma in 1960 that William Newcomb (1927–1999), a theoretical physicist at the Lawrence Radiation Laboratory, now the Lawrence Livermore National Laboratory (LLNL), in California, created an even more perplexing puzzle. Newcomb’s problem (now called Newcomb’s paradox) was formulated to help him explore the prisoner’s dilemma, and it is now generally believed that Newcomb’s paradox is a generalization containing the prisoner’s dilemma as a special case.

Curiously, Newcomb himself never published anything about his puzzle. Instead it first appeared in print in a 1969 paper by the Harvard philosopher Robert Nozick (1938–2002). The puzzle had been circulating via word-of-mouth in the academic community, but Nozick decided it needed a much wider audience. But what really brought Newcomb’s puzzle worldwide fame was when it appeared in the well-known popular math essayist Martin Gardner’s July 1973 “Mathematical Games” column of Scientific American (with a follow-up column in the March 1974 issue). So, here’s Newcomb’s paradox.

Imagine you are approached by an intelligent entity that has a finite but lengthy history of predicting human behavior with unfailing (so far) accuracy. It has, to date, never been wrong. You may think of this entity as (using Gardner’s examples) a “superior intelligence from another planet, or a super-computer capable of probing your brain and making highly accurate predictions about your decisions.” Or, if you like, you can think of the entity as God.¹ This entity makes the following presentation to you.

A week ago, the entity tells you, it predicted what you would do in the next few moments about the contents of those two mysterious boxes you’ve been wondering about that are sitting on a table in front of you. The boxes are labeled B1 and B2, and you can take either the contents of both boxes or the contents of box B2 only. The choice is entirely yours. B1 has a glass top, and you can see that the entity put $1,000 in that box. B2 has an opaque top, and you can’t see what, if anything, is in it. The entity, however, tells you that it put nothing in B2 if last week it predicted you would take the contents of both boxes, or it put $1,000,000 in B2 if last week it predicted you would take the contents of only B2.

What is your decision? Take both boxes, or just box B2 alone? The reason why this situation is called a paradox is because there are seemingly two quite different (but each clearly rational) ways to argue about what you should do. The two ways, however, lead to opposite conclusions!

25.2 DECISION PRINCIPLES IN CONFLICT

The first line of reasoning is similar to the one we used in prisoner’s dilemma, in that it is a dominance argument. As we did there, let’s make a table of the various potential outcomes as a function of what you decide and what the entity predicted:

Now, the entity (you reason) made its prediction a week ago and, based on that decision then, either did or didn’t put $1,000,000 in B2. Whatever it did is a done deal and can’t be changed by what you decide now. From the above table, it’s clear that you have the dominant strategy of taking the contents of both boxes, as $1,000 is greater than nothing (the entity predicted you’d take both boxes), and $1,001,000 is greater than $1,000,000 (the entity predicted you’d take only B2).

That makes sense to a lot of people, maybe you, too. But there is another, probabilistic argument that leads to the opposite conclusion. It goes like this. We don’t know that the entity is absolutely infallible. Yes, it’s true that it hasn’t been wrong yet, but its track record is finite. So, let’s say it has probability p of being correct and, since it has always been right up to now, it is almost certain that p is pretty close to 1 (but we don’t know that it is 1). So, for now it’s p. Now, in decision theory there is, besides the dominant strategy principle, another equally respected principle called the expected-utility strategy, in which you decide what to do by maximizing the expected utility that results from your choice. The utility of an outcome is simply the product of the probability of the outcome by the value of the outcome, and the expected utility is the sum of all the individual utilities.

Suppose you decide to take both boxes. The entity would have predicted (correctly) that you would do that with probability p, and with probability 1 − p it would have predicted (incorrectly) that you’d take only B2. So, the expected utility resulting from the choice of taking both boxes is

U_both = 1,000p + 1,001,000(1 − p) = 1,001,000 − 1,000,000p.

Next, suppose you decide to take only B2. The entity would have predicted (correctly) that you would do that with probability p, and with probability 1 − p it would have predicted (incorrectly) that you’d take both boxes. So, the expected utility resulting from the choice of taking only B2 is

U_B2 = 1,000,000p + 0(1 − p) = 1,000,000p.

Notice that as p → 1, we have U_both → 1,000 while U_B2 → 1,000,000.

The expected utility principle says you should decide to take only B2 if the entity is almost always correct. In fact, we can very loosely interpret what “almost” means since as long as p > 0.5005 (the entity simply flips an almost fair coin to make its prediction!) we have U_B2 > U_both, and the expected utility principle says you should take only B2.

I think you can now clearly see the paradox in Newcomb’s paradox. Two valid arguments, each eminent examples of rational reasoning, have led to exactly opposite conclusions. As Professor Nozick wrote in his 1969 paper,

I have put this problem to a large number of people, both friends and students in class. To almost everyone it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem with large numbers thinking that the opposite half is just being silly. Given two such compelling, opposing arguments, it will not do to rest content with one’s belief that one knows what to do. Nor will it do to just repeat one of the arguments loudly and slowly. One must also disarm the opposing argument; explain away its force while showing it due respect.

Well, logicians, philosophers, mathematicians, physicists, and just plain folks have been trying to do that over the more than forty years since Nozick wrote, and the noise and confusion continue to this day. What, you might wonder, did the creator of this puzzle think should be the choice? In a recent contribution,² the physicist Gregory Benford (who once shared an office with Newcomb at LLNL and often discussed the problem with him, long before it became famous) revealed that when he asked Newcomb that very question, the reply was a resigned “I would just take B2; why fight a God-like being?” I read that as meaning Newcomb, too, was as stumped by his own puzzle as everyone else!

This intellectual conundrum reminded Martin Gardner of one of the amusing little poetic jottings of the Danish scientist Piet Hein (1905–1996):

And what could be a better note than that on which to end a book of probability puzzles?

NOTES

1. If you think a problem that asks you to accept the possibility of such a mysterious entity is simply silly, that it poses a situation nobody with any serious intent would suggest (outside of theology, of course), you are wrong. In a brilliantly original book, the political scientist Steven Brams used two-person game theory to study the outcomes of an ordinary person interacting with an “opponent” that possess the attributes of omniscience, omnipotence, immortality, and incomprehensibility. That is, he studied the interactions of a human “playing against” what he called a “superior being”—or, in a theological setting, against God. See his book, Superior Beings: If They Exist, How Would We Know? (Springer-Verlag 1983).

2. David H. Wolpert and Gregory Benford, “The Lesson of Newcomb’s Paradox,” Synthese (Online First), March 16, 2011. There are a lot of references in this paper to the vast literature on the problem.