Monday, 16 March 2015

Why should you cooperate in the prisoners dilemma?

The prisoners dilemma is familiar to any student of game theory. One of its appealing properties is its seeming simplicity - there is an 'obvious' way to play the game. What is obvious in theory does not, however, always show up in reality. So, does that mean people are dumb or the theory is dumb? The conventional approach seems to be to say that people are dumb. But, I think its the theory that needs a rethink. Here is one reason why:
          The table below gives an example of the prisoners dilemma. Alice and Barney are two work colleagues who have to produce a project together. If they cooperate they will finish the project in one hour. If they do not cooperate it will take them 10 hours. If Barney cooperates and Alice defects then Alice gets away with doing nothing while Barney spends 15 hours. Vice versa if Alice cooperates and Barney defects.
   So, what should Alice do? If Barney cooperates then she does better to defect - she saves one hour. If Barney defects then she also does best to defect - she saves five hours. Either way Alice should defect. We get, therefore, our simple prediction. And if both Alice and Barney defect then they take 10 hours each rather than the one hour they would have spent if cooperating. This is the theory for a one-shot interaction. 
      What happens if they are going to do, say, 10 projects over the course of a year? On the last (tenth) project the incentive to defect remains. Moreover, nothing that happens in the ninth project will change the incentive to defect on the last project. That means there is an incentive to defect on the ninth project. This logic can be rolled back to the eighth project, seventh and so on. Standard theory predicts that Alice and Barney will defect from the first project onwards. So, where is the flaw in the logic?
      Suppose that Alice thinks Barney is a 'nice guy' who will cooperate on every project unless Alice has previously defected. On the last project Alice still has an incentive to defect. On the ninth project, though, things are already different. If Alice cooperates she will spend one hour on the current project and no time on the last. If she defects she will spend no time on the current project but 10 hours on the last. She does better to cooperate. This logic rolls all the way back to the first project. If, for instance, she cooperates on the first nine projects Alice will spend a total of nine hours working. If she defects on the first project she can expect a total of 90 hours work.
       The standard theory would reject this story on the basis that Alice should not expect Barney to be a nice guy. But, abundant experimental evidence shows that nice guys do exist. There is nothing, therefore, bizarre about Alice expecting Barney might be nice. And she only needs to put a sufficiently high probability, say 25%, on Barney being nice to justify cooperating. Indeed it may be that both Alice and Barney are 'nasty guys' but end up cooperating because they expect the other may be nice.
       The ideas sketched above are not new. Indeed Kreps and Wilson set out such ideas back in 1982 with a paper on 'Reputation and imperfect information'. According to Google this paper has a huge 3332 cites. I think it is fair to say, however, that the insights from that paper have not fed into conventional thinking on games like the prisoners dilemma. That's probably because of a reluctance by many to combine behavioural insight with game theory. So, don't trust a game theorist if he or she tells you it is always best to defect when playing the prisoners dilemma. Your intuition might be a safer guide.