Newcomb’s Paradox Revisited: A Simple Thought Experiment

Newcomb’s Paradox Revisited: A Simple Thought Experiment

In a Nutshell

Imagine that you are presented with two boxes. The first is transparent and contains $1,000. The second is opaque and contains either $1 million or nothing.

You now have a choice—to take both boxes or to take only the opaque box. Which should you do?

In one sense, the answer seems obvious. Take both. Whatever is inside the opaque box, you will be $1,000 better off. But there’s a complication.

The Predictor

Before you arrived, a very accurate “Predictor” tried to predict what you would do. If it predicted that you would take both boxes, it left the opaque box empty.

If it predicted that you would take only the opaque box, it put $1 million in that box.

The Predictor, it should be noted, has already made its decision—the money is either there or it is not. Nothing you do now can change that. It’s already happened. And the Predictor, as we’ve already mentioned, is usually right.

Now which box, or boxes, do you take? Interestingly, there are two apparently compelling arguments that lead to opposing answers.

This is Newcomb’s Paradox, named after the physicist William Newcomb and popularised by the philosopher Robert Nozick.

The case for taking both boxes

The argument for taking both boxes is wonderfully simple.

The opaque box already contains either $1 million or nothing. Now suppose it contains $1 million. If you take one box, in this case you receive $1 million. If you take both boxes, you receive $1 million plus a handy $1,000 bonus.

Now suppose the opaque box contains nothing. If you take one box, you now receive nothing. If you take both, however, you receive $1,000, the amount in the transparent box.

So whichever state of the world actually exists, taking both boxes leaves you $1,000 better off. At least that’s the logic of the two-boxers.

The case for taking one box

Suppose instead of applying this strict logic, you examine what actually happens to people who make the decision to take one box or two.

As we’ve already highlighted, the Predictor is very accurate. People who take both boxes therefore tend to discover that the opaque box is empty and they leave with just the money in the transparent box, the $1,000.

People who take only the opaque box tend instead to discover it contains $1 million and leave as millionaires.

Based on this evidence, which group would you now rather join?

There’s something that seems odd about a theory of rational choice which tells you to behave like the people who almost invariably end up with $1,000 rather than to behave like those who almost always end up with $1 million.

And yet the two-box argument hasn’t gone away. The money was placed in the box before you made your choice and nothing you do now can change that.

This is what makes Newcomb’s problem a paradox rather than merely a puzzle.

Reason against evidence?

What we have exposed here is a conflict between two approaches to decision-making.

The two-boxer concentrates on causation. Your current choice cannot cause the Predictor to change a decision already made. This points to what is known as causal decision theory.

The one-boxer focuses instead on evidence. Choosing one box is extremely strong evidence that the opaque box contains $1 million, but that’s what we have observed in practice. This is closer to what’s known as evidential decision theory.

Neither side needs to believe in time travel or backward causation.

The one-boxer isn’t claiming that choosing the opaque box magically puts the money there. The claim is simply that, in a world containing such an accurate Predictor, there’s a strong relation between whether you choose one box or two and the contents of the opaque box.

Even so, the Predictor can’t change anything now, so why leave the $1,000 on the table?

What sort of person are you?

There is a way of thinking about the problem which I find particularly interesting.

Picture yourself standing in front of the boxes. You ask yourself, “So what should I do now?”

Given that the Predictor is so good at judging you and forecasting your likely behaviour, based presumably on something about you, perhaps the more important question you should be asking is: “What sort of person, what sort of decision-maker, should I want to be?”

Look at it like this. Suppose you could decide beforehand to become a one-boxer. If the Predictor is convinced of this, it would presumably be aware of this and put $1 million in the opaque box.

That sounds promising.

But there’s a catch.

You can’t simply decide in advance to be a one-boxer and then, when the moment arrives, take both boxes. If that is your plan, you were never really a one-boxer at all. So how do you convince the Predictor that you are indeed a genuine one-boxer? Simply put, to be the sort of person whom the Predictor expects to take one box, you may actually have to take one box.

So which would I take?

I would take one box.

Here’s why.

The two-box argument at first sight seems totally rational. After all, when I make my choice, the contents of the opaque box are already fixed. Taking the transparent box as well as the opaque box can’t possibly remove money from the other one. So, if there are a million dollars there, why not take the extra thousand? The reason is the evidence before your eyes. As you watch a succession of people playing the game, you notice that it’s the one-boxers who are walking away with the $1 million prizes. The two-boxers are left with the contents of the transparent box.

After watching this happen often enough, there comes a point when I stop worrying about why the one-boxers are winning the big money and simply join them.

This is what makes Newcomb’s Paradox so challenging and so fascinating at the same time. Reason appears to tell us to take two boxes. The evidence seems to point to one.

In the end, I guess I would rather be an irrational but rich one-boxer than a perfectly rational but poor two-boxer!

My book, Twisted Logic: Puzzles, Paradoxes, and Big Questions, is now available.