Imperfect information facilitates the evolution of reciprocity

• We consider the case in which errors in behavior occur and information is imperfect.
• Imperfect information can encourage the evolution of direct reciprocity surprisingly.
• Payoff non-linearity makes richer scenarios in the evolution of cooperation.

The existence of cooperation demands explanation since cooperation is costly to the actor. Reciprocity has long been regarded as a potential explanatory mechanism for the existence of cooperation. Reciprocity is a mechanism wherein a cooperator responds to an opponent's behavior by switching his/her own behavior. Hence, a possible problematic case relevant to the theory of reciprocity evolution arises when the mechanism is such that the information regarding an opponent's behavior is imperfect. Although it has been confirmed also by previous theoretical studies that imperfect information interferes with the evolution of reciprocity, this argument is based on the assumption that there are no mistakes in behavior. And, a previous study presumed that it might be expected that when such mistakes occur, reciprocity can more readily evolve in the case of imperfect information than in the case of perfect information. The reason why the previous study considers so is that in the former case, reciprocators can miss defections incurred by other reciprocators' mistakes due to imperfect information, allowing cooperation to persist when such reciprocators meet. However, contrary to this expectation, the previous study has shown that even when mistakes occur, imperfect information interferes with the evolution of reciprocity. Nevertheless, the previous study assumed that payoffs are linear (i.e., that the effect of behavior is additive and there are no synergetic effects). In this study, we revisited the same problem but removed the assumption that payoffs are linear. We used evolutionarily stable strategy analysis to compare the condition for reciprocity to evolve when mistakes occur and information is imperfect with the condition for reciprocity to evolve when mistakes occur and information is perfect. Our study revealed that when payoffs are not linear, imperfect information can facilitate the evolution of reciprocity when mistakes occur; while when payoffs are linear, imperfect information disturbs the evolution of reciprocity even when mistakes occur. Imperfect information can encourage the evolution of cooperation.