Saturday, 28 April 2012

The Prisoners' Dilemma

The Prisoners' Dilemma has fascinated me ever since I came across it.  (And yes, the point is that there are two prisoners, so the apostrophe is in the right place.)

This lovely example of a real-life (OK, TV game show) Prisoners' Dilemma has been widely shared on Twitter and Facebook recently. If you haven't already seen it, watch this clip before you read any further.

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Ibrahim and Nick are playing a prisoners' dilemma.  Each can choose to "Split" or "Steal".  If they both Split, they share £13,600.  If one Steals and the other Splits, the stealer gets all the money.  If both Steal, neither gets a penny.

In this example, Nick persuades Ibrahim that he is going to Steal, but offers (in an unenforceable deal) to share his winnings with Ibrahim if Ibrahim Splits.  The video shows what happens.  But let's do some analysis.

Suppose Ibrahim believes Nick will Steal.  His only hope is the outside-the-show deal: there is no other way for Ibrahim to get any money.  So the only way Ibrahim can gain anything at all is to Split.

But think about Nick's position. What does he think Ibrahim will do?  If Ibrahim Splits, then Nick will share the money with him, either by Splitting or by Stealing and then honouring the word-of-mouth agreement.  (We assume that Nick is an honourable man who would not renege on that agreement.)  On the other hand, if Ibrahim Steals, Nick gets nothing if he also Steals, but if he Splits he might hope that Ibrahim will feel guilty and share his winnings as Nick would have done.  So Nick will always do no worse if he Splits than if he Steals, and he might do better.  So logically Nick's best choice is to Split.

Now go back to Ibrahim.  By the logic above he can deduce that Nick will gain nothing by Stealing, so if Nick is rational Ibrahim can safely Steal and get all the money.

But would you have the courage to follow your logic in that situation?

So for all Nick's cleverness, the Prisoners' Dilemma remains.

That's the problem with the Prisoners' Dilemma.  In a one-off PD, there is no way round it.  Of course, the work of people like Robert Axelrod and Martin Nowak shows just how wonderful the iterated Prisoners' Dilemma is as a mathematical tool for understanding co-operation, but that's another story.