If an iterated PD is going to be iterated exactly N times, for some known constant N, then there is another interesting fact. The Nash equilibrium is to always defect. That is easily proved by induction; one might as well defect on the last turn, since the opponent will not have a chance to punish the player. Therefore, both will defect on the last turn. Thus, the player might as well defect on the second-to-last turn, since the opponent will defect on the last no matter what is done, and so on. For cooperation to remain appealing, then, the future must be indeterminate for both players. One solution is to make the total number of turns N random. The shadow of the future must be indeterminably long.

Interesting, the way that such a small piece of extra knowledge can lead to such a big change in the strategy.

Incidentally, although the Southampton team could be seen to have cheated in their use of messaging between prisoners (some of their prisoners weren’t seeking their own best interests), all of the otherwise ‘best’ solutions used an implied messaging techinque. They all used cooperating on the first iteration as a message that they would continue to cooperate to other like minded prisoners.