  We will study a population of individuals playing the infinitely repeated
Prisoner's Dilemma under replicator dynamics. The population consists of three
kinds of individuals using the following reactive strategies: ALLD (individuals
which always defect), ATFT (almost tit-for-tat: individuals which almost always
repeat the opponent's last move) and G (generous individuals, which always
cooperate when the opponent cooperated in the last move and have a positive
probability $q$ of cooperating when they are defected). Our aim is studying in
a mathematically rigorous fashion the dynamics of a simplified version for the
computer experiment in [Nowak, Sigmund, Nature, 355, pp. 250--53, 1992]
involving 100 reactive strategies. We will see that as the generosity degree of
the G individuals varies, equilibria (rest points) of the dynamics appear or
disappear, and the dynamics changes accordingly. Not only we will prove that
the results of the experiment are true in our simplified version, but we will
have complete control on the existence or non-existence of the equilbria for
the dynamics for all possible values of the parameters, given that ATFT
individuals are close enough to TFT. For most values of the parameters the
dynamics will be completely determined.
