Scribble, scribble, scribble

signs and wonders

On a recent bus journey, I passed two churches with signs outside them. One said:

Whatever your place
Tend it with care
God put you there

and the other said:

Reason is the worst enemy faith has

I am not, in general, an atheist of the hostile or crusading sort. But sometimes I do get rather cross.

newcomb

I discovered a little while ago that at least one intelligent and well-informed person who sometimes reads this stuff had never come across Newcomb's paradox before. It's worth knowing about, if only because it will make your head spin.

Imagine that there is some being (call it Bill) with a well-established ability to predict people's behaviour. In particular, Bill has demonstrated to your satisfaction that he can predict, hours in advance and very reliably (let's say well over 99%), what you will do in a wide variety of situations, and he has demonstrated a similar ability to predict other people's behaviour in the exact situation you're about to be placed in, which is this:

In front of you are two boxes. One is made of glass; you can see into it. It contains a cheque for £1000. The other is made of steel, and welded shut; you can't tell what's in it. Bill has put in it either a blank piece of paper or a cheque for £1000000. You may take (and keep) both boxes, or just the opaque one. But here's the rub: Bill has used his pred1cti0n sk1llz and has put the big cheque in the steel box if he thinks you'll take only that box, and the blank piece of paper if he thinks you'll take both. What do you do?

Let me recap, in two different ways suggestive of two different answers.

• There is almost certainly £1000000 in the steel box if you choose to take only that box, and almost certainly nothing of value in it if you choose to take both boxes. (Therefore: Take only the steel box. Obviously.)
• There may or may not be £1000000 in the steel box, but there is certainly £1000 in the glass box, and whatever is in the steel box there is therefore £1000 more in the two boxes together than in the steel box alone. (Therefore: Take both boxes. Obviously.)

Almost everyone finds it entirely obvious what it's rational to do in this peculiar situation. Unfortunately, there is no agreement on which of the two courses of action is the obviously rational one.

When we ask questions like "What should I do in this situation?", I think we are implicitly operating with a possibly-naïve notion of how the world works, and if Bill's predictive skills are possible then it's definitely too naïve. That notion is as follows: There are various ways the world (past and present) could be, all of which look just like the actual world in the past, and which differ in the future according to your choice; all the future differences are consequences of your choice and could in principle be traced back to that choice itself, via causes flowing forward in time.

Generally, this view of things works well. But in a world containing Bill, it breaks down: your future choices are strongly correlated somehow with things in the past, and by Bill's clairvoyant or simulatory prowess those things in the past are in turn strongly correlated with something else contemporary with (even preceding) your choice, with macroscopic consequences. Thus, our key assumption – that all the differences between the hypothetical futures of the world in which you chose differently flow from that choice – fails.

(Very likely it fails in the real world too, but so far as we can tell it generally fails benignly, or at least in ways we don't see because our powers of perception and prediction are unlike Bill's.)

But that assumption is a fundamental part of what we mean by asking questions like "What should you do?". It's hard to answer this question when it concerns Newcomb's situation because one of the question's presuppositions is false. It's like asking what colour electrons are.

You may notice that the question I actually asked was "What do you do?". Not being Bill, of course, I don't really know, even if "you" means me. But I'm pretty sure that I take only the steel box. Is that a rational decision? I think the question has little meaning. But it's rational at least in this sense: making that decision is strongly predictive of getting £1000000 instead of £1000.