The occasional absence of resources for cooperation and its role in the evolution of direct reciprocity

Reciprocity is regarded as a mechanism that possibly explains the evolution of cooperation in repeated encounters. Occasionally, even if individuals want to cooperate, they can lack the necessary resources for providing help to others, thereby preventing them from the engagement in benevolent interactions. Unlike previous investigations, the present study examines the situation where a player sometimes knows whether the opponent player had resources for cooperation or not. Using an evolutionary stable strategy (ESS) analysis to obtain the memory-one strategy whose stability condition against the invasion by unconditional defectors is the loosest, we find that forgiveness does not influence the evolutionary outcomes whereas persistence (whereby players imitate their own behavior when knowing that the opponents did not have resources for cooperation and defected) facilitates the evolution of cooperation. Forgiveness in our model means that a player decides to cooperate if the opponent did not have the resources for cooperation and defected previously. In addition, we introduce lying strategists who can pretend having no resources for cooperation, and we analyze the three-strategy game played by lying strategists, the honest naive strategists, and unconditional defectors, finding that at low cost-to-benefit ratios, unconditional defection and the lying strategy can be stable, while the honest naive strategists diminish. Our results highlight the importance of accessibility of information about opponent's resources for cooperation and its effects on the evolutionary dynamics of direct reciprocity.