Evolutionarily Stable Strategy - Prisoner's Dilemma and ESS

Prisoner's Dilemma and ESS

A common model of altruism and social cooperation is the Prisoner's dilemma. Here a group of players would collectively be better off if they could play Cooperate, but since Defect fares better each individual player has an incentive to play Defect. One solution to this problem is to introduce the possibility of retaliation by having individuals play the game repeatedly against the same player. In the so-called iterated Prisoner's dilemma, the same two individuals play the prisoner's dilemma over and over. While the Prisoner's dilemma has only two strategies (Cooperate and Defect), the iterated Prisoner's dilemma has a huge number of possible strategies. Since an individual can have different contingency plan for each history and the game may be repeated an indefinite number of times, there may in fact be an infinite number of such contingency plans.

Three simple contingency plans which have received substantial attention are Always Defect, Always Cooperate, and Tit for Tat. The first two strategies do the same thing regardless of the other player's actions, while the later responds on the next round by doing what was done to it on the previous round—it responds to Cooperate with Cooperate and Defect with Defect.

If the entire population plays Tit-for-Tat and a mutant arises who plays Always Defect, Tit-for-Tat will outperform Always Defect. If the population of the mutant becomes too large — the percentage of the mutant will be kept small. Tit for Tat is therefore an ESS, with respect to only these two strategies. On the other hand, an island of Always Defect players will be stable against the invasion of a few Tit-for-Tat players, but not against a large number of them. If we introduce Always Cooperate, a population of Tit-for-Tat is no longer an ESS. Since a population of Tit-for-Tat players always cooperates, the strategy Always Cooperate behaves identically in this population. As a result, a mutant who plays Always Cooperate will not be eliminated. However, even though a population of Always Cooperate and Tit-for-Tat can coexist, if there is a small percentage of the population that is Always Defect, the selective pressure is against Always Cooperate, and in favour of Tit-for-Tat. This is due to the lower payoffs of cooperating than those of defecting in case the opponent defects.

This demonstrates the difficulties in applying the formal definition of an ESS to games with large strategy spaces, and has motivated some to consider alternatives.