Cooperation is less likely to evolve in a finite population with a highly skewed distribution of family size

In the context of the finitely repeated Prisoner's Dilemma with the possibility of cooperating or defecting each time, the strategy tit-for-tat (TFT) consists in cooperating the first time and copying the strategy previously used by the opponent the next times. Assuming random pairwise interactions in a finite population of always defecting individuals, TFT can be favoured by selection to go to fixation following its introduction as a mutant strategy. We deduce the condition for this to be the case under weak selection in the framework of a general reproduction scheme in discrete time. In fact, we show when and why the one-third rule for the evolution of cooperation holds, and how it extends to a more general rule. The condition turns out to be more stringent when the numbers of descendants left by the individuals from one time-step to the next may substantially differ. This suggests that the evolution of cooperation is made more difficult in populations with a highly skewed distribution of family size. This is illustrated by two examples.