Evolutionary Stability in Relation to Populations in Nature
Nowak, Sasaki, Taylor, and Fudenberg argue in this article that a single individual using a cooperation strategy such as “tit-for-tat” (TFT) can infiltrate a finite population of individuals who always defect (A||D) in a repeated prisoner’s dilemma type situation. In TFT, an individual cooperates the first round and then does whatever the other player did the round before after that. In A||D, an individual always chooses to not cooperate. Usually it is assumed that TFT cannot invade an infinitely large population of defectors in a game of finite length because the payoff for the individuals using TFT is smaller than that of the individuals using A||D, unless the percentage of individuals using TFT is above some critical value (called the invasion barrier). If two strategies are the best response to themselves, then cooperation will invade a population of defectors if one of two conditions holds. One, if the payoff of cooperating when the other player does not is higher than the payoff of not cooperating when the other player does, or two, if the starting percentage of cooperating individuals is larger than the invasion barrier, as stated earlier.
This relates very closely to what we learned in class about game dynamics and evolutionary stability. A||D is evolutionary stable because the payoff for cooperating if only a small x percent of the population is cooperating is lower than that for not cooperating. TFT is also evolutionary stable because if only a small x percent of the population is not cooperating, then the payoff for cooperating is higher than that for not cooperating. This situation also relates to what we learned about Nash Equilibria earlier in the semester. The best response to each strategy (cooperating and not cooperating) is the strategy itself. The Nash Equilibria are not necessarily stable, though. If not enough of the population uses the TFT strategy (that is, if it does not exceed the invasion barrier), then TFT will not invade the population. If the percent of the population using the TFT strategy exactly equals the invasion barrier, the population is at an unstable equilibrium, because while the percent of the population using the strategy will remain constant, if the percent were to decrease at all, the TFT strategy would disappear.
Link: http://www.nature.com/nature/journal/v428/n6983/full/nature02414.html