The Predictor Problem: Should you open one box or two?

Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

You win a quiz show and are offered a choice. You are presented with a transparent box containing £10,000 and an opaque box which contains either £100,000 or nothing. Now, you can open the opaque box and take what is inside, or you can open both boxes and take the contents inside both. Which should you choose? Well, if that’s all the information you have, it’s obvious that you should open both boxes. You certainly will not win less than by just opening one of the boxes, but you might win a lot more. So far, so good. But now introduce an additional factor. Before making your decision, you had to undergo a computerised sophisticated psychometric test (a Predictor) which you are now told has been unerring in its prediction of what hundreds of previous contestants would decide. Whenever they chose both boxes, there was nothing inside the opaque box. Whenever they had chosen just the opaque box, however, they found a cool £100,000 inside. When you make your decision the computer’s decision has already been made. The contents of the opaque box have already been placed there. What is happening is that the Predictor informs the game show organisers its prediction of whether a contestant will choose two boxes or one box. Whenever it predicts that the contestant will choose two boxes, no money is placed in the opaque box. Whenever it predicts that the contestant will choose just the opaque box, £100,000 has been deposited in the box.

This is essentially the basis of what is known as Newcomb’s Paradox or Newcomb’s Problem, a thought experiment devised by William Newcomb of the University of California and popularised by philosopher Roberz Nozick in a paper published in 1969.

So what should you do? Open just the opaque box or open both boxes.

In his paper, Nozick writes that “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 opposing half is just being silly.”

The argument of those who argue for opening both boxes (the so-called ‘two-boxers’) is that the money has already has already been deposited at the time you are asked to make your decision. Taking two boxes can’t change that, so that’s the rational thing to do.

The argument of those who argue for opening just the opaque box (the so-called ‘one-boxers’) is that the psychometric test is either a perfect or near-perfect predictor of what you will do. It has never got it wrong before. Every single previous contestant who has opened two boxes has found the opaque box empty, and every single previous contestant who has opened just the opaque box has won the £100,000. So do what all the evidence tells you is the sensible thing to do and open just the opaque box.

One way of considering the question is to ask whether your choice in some way determines the choice of the Predictor, and thereby the decision as to whether to place the £100,000 in the box. Well, there’s no time-travelling retro-causality involved. The predictor is basically a piece of computer software which bases its prediction on a psychometric test. It just so happens that the test is uncannily accurate in knowing what people will do.

Look at it this way. Bottom line is that you have a free choice, so why not open both boxes? The problem is that if you are the type of person who is a two-boxer, the predictor will have found this out from the super-efficient psychometric test. If you are the type of person, however, who is a one-boxer, the predictor will find that out too.

So it’s not that there is any good reason in itself to open one box rather than two. After all, what you decide now can’t change what is already in the box. But there is a good reason why you should be the type of person who only opens one box. And the best way to be the sort of person who only opens one box is to only open one box. For that reason, the way to win the £100,000 is to agree to open just the opaque box and leave the other box untouched.

But why leave behind that extra £10,000 when the £100,000 which you are about to win is already in the box?

That’s Newcomb’s Paradox. You decide! Are you are a one-boxer or two?