This is a classic puzzle which, like all such, comes in the form of a story. Here is one version:
Alice and Bob are criminals. No question. They have been caught red-handed in a botched robbery of the Smalltown Store, and are now in jail awaiting trial.
The police have realised that Alice and Bob match the description of the pair who successfully robbed the Bigtown Bank last month. They really want to get a conviction for that, but with no evidence apart from the resemblance they need to get a confession.
So they say to Alice: “Look, you are going to get a 1 year sentence for the Smalltown Store job, no question. But if you co-operate with us by confessing that the two of you did the Bigtown Bank heist then we’ll let you go completely free. You can claim Bob was the ringleader and he’ll get a 10 year sentence.”
Alice thinks a moment and asks two questions.
“Are you making the same offer to Bob? What happens if we both confess?”
The police tell her that yes, they are making the same offer to both of them. And if both confess, they’ll get 6 years each.
OK, that’s the story. All that circumstantial detail is just to lead up to this decision table, which Alice is now looking at:
That’s the problem in a nutshell. Before we look at it there are maybe a few points to clear up
- Alice and Bob are not an item. They are just business partners. Each is aiming to minimise their own jail term, and what happens to the other is irrelevant for them.
- ‘Go free’ really does mean that – there are no vengeful families or gang members to bring retribution on an informer.
- Whether they actually committed the Bigtown Bank job is completely irrelevant to the puzzle.
OK, let’s get back to Alice. She reasons as follows:
“I don’t know what Bob is going to do. Suppose he denies the bank job. Then I should confess, to reduce my sentence from 1 year to zero. But what if he confesses? In that case, I’d better confess too, to get 6 years rather than 10. Whichever choice Bob makes, the better option for me is to confess. So I’ll confess.”
Bob will, of course, reason the same way. If Alice denies, he should confess. If Alice confesses, he should confess. 0 is less than 1, and 6 is less than 10. Therefore he should confess.
The logic is irrefutable. But look at that table again. The prisoners have firmly chosen the top left box, and will both serve 6 years. That’s a terrible result! It’s not only the worst total sentence (12 years), its the next-to-worst individual sentence (6 years is better than 10, but much worse than 0 or 1). Clearly the bottom right is the box to go for. It’s the optimal joint result and the next-to-optimal individual result.
That is obvious to us because we look at the table as a whole. But Bob (or Alice) can only consider their slices through it and either slice leads to the Confess choice. To see it holistically one has to change the question from the Prisoner’s Dilemma to the Prisoners’ Dilemma. That’s only the movement of an apostrophe, but it’s a total readjustment of the viewpoint. A joint Bob+Alice entity, if the police put them in one room together for a couple of minutes (but they won’t), can take the obvious bottom-right 1+1 choice. Separate individual Bob or Alice units, no matter how rational, cannot do that.
This is what the philosophers call emergence. The whole is more than just the sum of its parts. A forest is more than a number of trees. An animal is more than a bunch of cells. It’s generally discussed in terms of complex large-N systems: what’s nice about the Prisoner’s Dilemma is that emergence appears with just N=2. There is a Bob+Alice entity which is more than Bob and Alice seperately, and makes different (and better) decisions.
There’s also a lesson for politics. It’s an illustration of the way that Mrs Thatcher was wrong: there is such a thing as society, and it is more than just all its individual members. Once you start looking for them, the world is full of examples where groups can do things that individuals can’t – not just from the “united we stand” bundle-of-sticks argument but because they give a different viewpoint.
- I should stockpile lavatory paper in case there’s a shortage caused by people stockpiling lavatory paper.
- When recruiting skilled workers it’s quicker and cheaper for me to poach yours rather than train my own.
- My best fishing strategy is to catch all the fish in the pond, even though that leaves none for you, and none for me tomorrow.
- If I get another cow that will always give me more milk, even though the common grazing we share is finite.
Following the last instance, economists call this “The tragedy of the commons”. It’s the point at which Adam Smith’s “invisible hand” fails.
This tells us something about democracy. A society or a nation is more than just the individuals that make it up. E pluribus unum means that something larger, more powerful and – dare one say it – better can emerge. So democracy is more than just arithmetical counting noses, democracy provides the means whereby men and women can speak with one voice as a distinct people. That’s the ideal, anyway, and – even if the form we’ve got is clunky and imperfect – some of us still try to believe in it.