Can selfish players learn to cooperate?

In theory, yes, as long as they never know which round is the last. Played once, the prisoner's dilemma rewards betrayal: defecting always pays better than cooperating, whatever the other player does. Repetition alone does not fix that. If both players know the game lasts a fixed number of rounds, defecting in every round is still the dominant strategy: one might as well defect on the last turn, since the opponent will not have a chance to retaliate, and the same reasoning then unravels every earlier round. For cooperation to emerge between rational players, the number of rounds must be unknown or infinite. Robert Aumann showed in 1959 that rational players interacting in indefinitely long games can sustain cooperation.

Robert Axelrod's computer tournament, reported in his 1984 book The Evolution of Cooperation, put the idea to the test with programs that varied widely in complexity, initial hostility and capacity for forgiveness. The simplest, tit for tat, won: it cooperates first, then copies the opponent's previous move. After analysing the top scorers, Axelrod stated conditions for success. Besides being nice and forgiving, a winner must retaliate, because always cooperating is a very bad choice that nasty strategies frequently exploit. And it must not be envious: it does not strive to score more than its opponent.

The rules around the game matter too. In a 2004 tournament, the University of Southampton submitted 60 programs designed to recognise each other during the first five to ten moves. Once they had, one always cooperated and the other always defected, handing the defector the maximum points. Southampton took the first three places, by taking advantage of the fact that multiple entries were allowed.

So tit for tat is not invincible: it is not always the winner of a given tournament, only better than its rivals over a series of them.