Labyrinths of Reason (40 page)

Newcomb devised his paradox while pondering the “prisoner’s dilemma,” another well-known situation in game theory. It is worth describing briefly that situation too.

In the prisoner’s dilemma, two wrongdoers are arrested for a crime. The police interrogate the prisoners separately so that they may not collaborate on a lie. Each prisoner is offered a deal. The corrupt police want a scapegoat. If the prisoner confesses everything, they’ll let him go (provided the partner doesn’t do the same thing). Each prisoner must make his decision without consulting the other, and knowing that the other is being offered the same deal. What is the best course for a prisoner to take?

The best individual outcome for either prisoner results when he confesses and his partner doesn’t. That lets him off the hook completely. Conversely, the
worst
fate is to be the one who doesn’t confess. Given the partner’s testimony, the judge is sure to throw the book at the guilty party who persists in his he.

Things are almost as bad if both confess. Then both will be convicted. Still, neither is quite as bad off as in the case where his partner gets off scot-free: The fury of the law is split between the two. Likewise, things are fairly good for both parties if both don’t confess. The police will still suspect the pair, but they may not have enough evidence to secure a conviction.

The prisoner’s dilemma explores the conflict between the good of the individual and the good of all. The prisoners really shouldn’t confess, because that is best for the pair. But assuming the other guy won’t confess, each prisoner is tempted to better his own situation by turning state’s evidence. The real-world versions of this are so numerous and obvious they needn’t be enumerated.

As you will gather, the prisoner’s dilemma is closely related to chicken. In both the participants are tempted to do something that would be disastrous if both did it (not swerve, turn state’s evidence). Call this course of action “defecting.” In chicken the worst possible
outcome results when both players defect. In the prisoner’s dilemma, the worst outcome results when the other guy defects and you don’t. Thus the temptation to defect is more acute in the prisoner’s dilemma. In chicken, if you know your opponent is going to defect (by being omniscient, say), you can only grit your teeth and
not
defect. In the prisoner’s dilemma, knowing your partner will defect is all the more reason for you to defect.

Newcomb’s paradox goes like this: A psychic claims to have the ability to predict your thoughts and actions days in advance. Like most psychics, he does not claim total accuracy. He is right about 90 percent of the time. You have agreed to take part in an unusual test of the psychic’s powers. A TV news program has provided the facilities and put up a large sum of money; all you have to do is abide by the conditions of the experiment.

On a table in front of you are two boxes: A and B.

Box A contains a thousand-dollar bill. Box B either contains a million dollars or is empty. You cannot see inside it. Of your own free will (if there is such a thing), you must choose either to take box B only or to take both boxes. Those are the only options.

The catch is this: Twenty-four hours ago, the psychic predicted what you would choose. He decided whether to put the million dollars in box B. If he predicted that you would take
only
box B, he put the million dollars in it. If he foresaw your taking both boxes, he left box B empty.

Personally, you don’t care whether the psychic’s powers are proven or discredited. Your only motive is leaving the experiment with as much money as possible. You are not so wealthy that money means nothing. The thousand dollars in box A is a lot of money to you. The million dollars is a fortune.

The conditions of the test have been and will be enforced scrupulously. You need entertain no doubts that box A contains the $1000. Box B can contain only $1 million or nothing at all, based on the psychic’s prediction. No one is trying to trick you on that score. A trusted friend was present at the time the psychic made his prediction, and made sure he obeyed the rules about putting the money in the boxes.

Just as certainly,
you
will be prevented from circumventing the rules. Armed guards will prevent nihilist acts like not taking
any
box. Nor can you cheat the psychic by basing your decision on
something other than your own mental processes. You can’t decide on the basis of a coin toss or whether the number of shares traded that day is odd or even. You have to analyze the situation and decide on the most profitable of the two options. Of course, the psychic has anticipated your analysis. What should you do—take both boxes or just B?

One reaction to the paradox goes like this: Psychics! Everyone knows that’s a lot of hooey! So all that business about the “prediction” is irrelevant. It boils down to this: There are two boxes, they might contain money, and you’re free to take them.

It’s silly to take just box B when there is a guaranteed $1000 in box A. That is exactly like not picking up a thousand-dollar bill you see on the street. The contents (if any) of box B are not going to vanish if you take both boxes. No one, including the psychic, claims that that sort of telekinetic power is at work. The boxes were hermetically sealed twenty-four hours ago. You should take both.

There is also a strong argument for taking only box B. Remember, the psychic
is
usually right. That is a given. Chances are, he would be right about your taking both boxes, in which case you’d get a measly thousand. Meanwhile, a sucker taken in by his claims would get a million.

What if the experiment has been conducted hundreds of times previously, with the psychic almost always being right? This should not change things, since the psychic’s accuracy is postulated. Bookmakers are accepting side bets on the outcome of the experiment. Provided you take box B only, they quote odds of 9 to 1 in favor of its containing a million dollars. If you take both boxes, your chance of getting a million is a long shot at 9 to 1 against. The bookmakers are not quoting these odds out of altruism. These are the actual probabilities, as near as anyone can determine them.

Money being the only thing that matters in the experiment, the case for taking box B can be stated in dollars and cents. If you take both boxes, you will get a sure $1000 (box A) and further stand a 10 percent chance at a million dollars—the latter if the psychic wrongly predicted that you would take box B only. On the average, a 10 percent chance of $1 million is worth $100,000. The total expected profit from taking both boxes is therefore $1000 + $100,000 or $101,000.

If instead you take box B only, you stand a 90 percent chance
that the psychic will be right and leave a million dollars in there. That’s worth $900,000 on the average. It is strongly in your favor to take B only. The better the psychic’s batting average, the more you profit by taking only box B. If he is right 99 percent of the time, the strategies would be worth $11,000 (both boxes) and $990,000 (B only). In the limiting case where he’s
always
right, it amounts to a choice between $1000 (both boxes) or $1 million (B only).

No one has yet squared these opposing viewpoints to everyone’s satisfaction. The ingenuity of suggested resolutions to Newcomb’s paradox may be without parallel. Among the bizarre explanations that have been seriously offered are that the sealed boxes constitute a Schrödinger’s cat-like situation and are neither empty nor full until opened!

The conventional analysis of the prisoner’s dilemma is wanting. Notice the parallels. As with the prisoners, you and the psychic really should “cooperate” by predicting, and then taking, box B only. But assuming that the psychic did cooperate, you are sorely tempted to enrich yourself further by taking both boxes. It is a contention of game theory that one should never be the first to defect in a prisoner’s dilemma situation. But how can that advice apply here? The psychic has played his hand, and there are no future consequences to worry about.

Many variations on the basic situation have been proposed in the attempt to make the correct course clearer. The predictor can be a being from another planet, God, a spouse of twenty years who “knows how you think,” or a computer that has been programmed with extensive information on the state of all the neurons in your brain. You can vary the predictor’s accuracy from 50 to 100 percent to see what difference it makes. Some variations stack the deck in favor of one choice, but none exorcises the paradox.

The paradox depends on faith in the predictor’s abilities. Suppose the “psychic” has no predictive power whatsoever and just tosses a coin to decide whether to put the million in box B. In that case, everyone must agree that you should take both boxes. Whether the predictor is right or wrong, you’re $1000 ahead for taking both boxes. Figuring the odds leads you to the same conclusion. Taking both boxes gets you a sure $1000 plus a 50 percent chance at $1 million ($501,000 total) vs. 50 percent of $1 million ($500,000) for taking just box B.

The paradox further requires that the predictor’s accuracy be great enough to make up for the loss of the sure thing in box A. The accuracy must be greater than 50.05 percent with the dollar amounts stated. In general, the accuracy must be at least
(A + B)/
2B
, where
A
is the amount in box A, and
B
is the amount that may or may not be in box B.

The argument for taking both boxes is more trenchant if box A is made of glass and box B has a glass window on its far side. You can verify for yourself that the $1000 is in box A. A nun who has taken a vow of truthfulness sits on the opposite side of the table and can see in box B’s window. The nun is not allowed to betray the contents of box B by facial expressions or any other means, but after the experiment is over she will be able to affirm that the money didn’t vanish or appear out of thin air as you made your choice. With this setup, wouldn’t you feel silly just taking box B? The psychic has already committed himself. The nun will see you either passing up a sure $1000 and taking an empty box B—then you’d
really
feel stupid—or getting the million but still passing up the thousand dollars for no reason whatsoever.

Prior to the experiment, you announce that 10 percent of your proceeds will be donated to an orphanage. The nun, who sees what’s in the boxes, silently prays that you will do whatever will result in the greater donation.
There can be no doubt what the nun wants you to do
. She wants you to take both boxes. No matter what she sees, your taking both boxes means an extra $100 for those needy waifs.

In still another variation, this one suggested by Newcomb, both boxes are glass on all sides. Box B contains a slip of paper on which is written a very large odd integer. The experiment’s sponsors have agreed to pay the bearer of that slip of paper $1 million if the number is prime. The psychic has chosen the number so that it will be prime only if he predicted you would take only box B. You can see the number and make a record of it for reference, but you are not allowed to determine if it is prime until you have made your choice. Now, certainly a mathematical fact is not going to change. Long before the stars existed, there was number; nothing you do here and now on this insignificant planet will make any difference in the realm of mathematics. This version of the paradox is one last chance to cast out any doubts you may have about your decision affecting the prediction through some kind of weird backward causality.

Like an automobile’s chronic knock, the paradox lingers, even
when you start to disassemble it. Suppose that the experimenters elect to sweeten the odds in your favor. Under modified rules, you are allowed—in fact, encouraged—to open box B first and
then
decide if you want to take box A as well. After you open B and see what’s in it, you can clutch the million (if it’s there) tightly to your bosom, and even deposit it in your bank account if you still harbor some silly idea that the money may yet go
poof!
and vanish. Then,
only then
, must you decide whether to take the $1000 in box A as well.

Are we not all agreed that you would be stark, raving, foaming-at-the-mouth mad
not
to take A? You’d certainly take it if you found box B empty. It would be equally irrational not to take A after finding the million in B and banking it.

That agreed, not everyone is rational. There would still be an occasional moron who would open box B, find a million dollars, and not take box A. Of course, everyone with half a brain opens B, finds nothing, and takes box A.

Prediction of human behavior brings up questions of free will. You can excise free will from Newcomb’s paradox thus: The psychic is not what he claims. He has no powers of clairvoyance. Instead, he has a device that causes the subjects to choose whichever option he selects. The psychic decides you’ll take both boxes, fixes them accordingly,
and then pushes a button that makes you take both boxes
.

This eliminates concern that the prediction in Newcomb’s paradox is physically impossible. Of course, we can no longer ask, “What would you do?” You’d do whatever the psychic decided you’d do. The most we can ask is, “Who would you rather be, one of the zombies who get $1000 or one of the zombies who get $1 million?” Well, of course, you’d rather get the million. Some
real
money is the least you deserve, giving up your free will like that.

If you agree with that, does it really matter
how
the psychic achieves his accuracy, through prediction or mind control? You’re concerned only with money, not with making some existential statement. Shouldn’t you then take box B even if there is no mind control and you do have free will?

Opinion remains strongly divided on Newcomb’s paradox. There is this distinction between the both-boxes and the box-B camp: Everyone who would take box B only does so in the expectation of getting the $1 million. The both-boxes people are divided among those who austerely expect to get only the $1000 and those who imagine they have a greater than stated shot at the $1 million too.