Supplementary Information from Stabilizing unconditional cooperation

The ultimate goal of research on the evolution of cooperation could be conceived as finding a method for stabilizing strategies that always cooperate, that never deviate from cooperation, that never exploit, and never retaliate, because all such activities are inherently problematic. The hope of achieving this cooperative utopia seems unjustified, especially in the context of direct reciprocity, which relies on the principle that cooperative partner strategies succeed in defending themselves by using moderate retaliation. Here, we propose a dynamic that goes a certain way towards achieving the desirable goal. In the ground state, the population consists of individuals that use always-cooperate (ALLC). Occasionally defectors, in the form of always-defect (ALLD), invade the population. Their presence triggers a mutation from ALLC to another strategy, X, with the aim to avert the take-over of defectors. In the absence of X, ALLD dominates ALLC, but in the presence of X, the invasion attempt might fail and subsequently the ALLC ground state can be restored. We study this mutation-selection process in finite and infinite populations. We identify the properties of the ideal rescue strategy. We derive an optimum mutation rate that maximally stabilizes ALLC.