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. It contains either $1 million or nothing.

You have a choice. You can take both boxes, or you can take only the opaque box.

Which should you do?

So far the answer seems obvious. Take both. Whatever is inside the opaque box, you will be $1,000 better off.

Unfortunately, there is a complication.

The Predictor

Before you arrived, a remarkably 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 it.

The Predictor has already made its decision. The money is either there or it isn’t. Nothing you do now can change what happened earlier.

There is just one more piece of information. The Predictor is almost always right.

Now which box, or boxes, do you take?

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

It has the unusual property that two apparently compelling arguments lead to opposite answers.

Take both boxes

The argument for taking both boxes is wonderfully simple.

The opaque box already contains either $1 million or nothing.

Suppose it contains $1 million. If you take one box, you receive $1 million. If you take both, you receive $1 million plus a handy $1,000 bonus. 

Now suppose the opaque box is empty. If you take one box, you now receive nothing. If you take both, you receive $1,000. 

So whichever state of the world actually exists, taking both boxes leaves you $1,000 better off.

What more is there to say?

Quite a lot, according to the one-boxers.

Take one box

Suppose you look not at the logic of the individual decision but at what actually happens to people who face it.

The Predictor is extraordinarily accurate. People who take both boxes therefore tend to discover that the opaque box is empty and leave with $1,000.

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

If that is the evidence, which group would you rather join?

There is something decidedly 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 those who almost always end up with $1 million. 

And yet the two-box argument hasn’t disappeared. The money was placed in the box before you chose. Reaching for the transparent box cannot somehow reach backwards through time and remove the million dollars.

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

Reason against evidence?

One way of describing the disagreement is as a conflict between two approaches to decision-making.

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

The one-boxer concentrates instead on evidence. Choosing one box is extremely strong evidence that the opaque box contains $1 million. This is closer to 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, your choice and the contents of the box are very strongly related.

That distinction matters.

But it doesn’t make the uncomfortable question go away.

If the Predictor has already done its work, why leave the $1,000 on the table?

What sort of person are you?

There is another 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 forecasting your behaviour from your character, reasoning or disposition, perhaps the more important question you should be asking is: is: “What sort of decision-maker should I want to be?”

Look at it like this. Suppose you could decide beforehand to become a committed one-boxer. The Predictor would presumably anticipate this and put $1 million in the opaque box.

That sounds promising.

But there is a catch.

You cannot simply decide in advance to be a one-boxer in order to fool the Predictor and then, when the moment arrives, take both boxes. If that is your plan, you aren’t really a one-boxer at all. And a sufficiently good Predictor will presumably know it.

To be the sort of person whom the Predictor expects to take one box, you may actually have to be the sort of person who does decide to take one box.

The distinction between choosing an action and choosing the kind of person who performs that action begins to get rather blurred.

So which would I take?

I suspect I would take one box.

Here’s why. 

The two-box argument at first sight does seem impeccable. At the instant I make my choice, the contents of the opaque box are fixed. Taking the transparent box as well cannot possibly remove money from the other one. If there is a million dollars there, why wouldn’t I take the extra thousand?

But then I imagine watching a succession of people play the game.

The two-boxers confidently explain the dominance principle and walk away with $1,000. But it’s the one-boxers I’m seeing walk away with $1 million.

After watching this happen often enough, I suspect I would stop worrying about why the one-boxers were richer and join them.

Perhaps that is what makes Newcomb’s Paradox so irritating. Reason appears to tell us to take two boxes. The evidence appears to tell us to take one.

And I would rather be an inexplicably rich one-boxer than a perfectly rational two-boxer with $1,000.

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