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.合作演化研究的终极目标，可被理解为探寻一种能够稳定特定策略的方法：这类策略始终保持合作、绝不偏离合作轨道、从不利用他人、亦从不实施报复——因为上述各类行为本身均存在固有缺陷。实现这一合作乌托邦的愿景似乎难以成立，尤其是在直接互惠（direct reciprocity）的情境下，其核心原则为合作型伙伴策略需通过适度报复实现自我防卫。对此，本文提出一种动态机制，可在一定程度上助力达成这一理想目标。初始基态下，种群由始终合作（always-cooperate, ALLC）策略持有者构成。偶尔会有以始终背叛（always-defect, ALLD）策略为代表的背叛者入侵该种群。背叛者的存在会触发ALLC策略向另一策略X的突变，其目的是阻止背叛者接管种群。若无策略X存在，ALLD将主导ALLC种群；但当策略X存在时，背叛者的入侵尝试可能失败，进而可恢复ALLC的初始基态。本文针对有限种群与无限种群场景下的这一突变-选择过程展开研究，明确了理想救援策略的各项特性，并推导得出可使ALLC策略实现最大化稳定的最优突变率。