Withhold-judgment and punishment promote cooperation in indirect reciprocity under incomplete information

Indirect reciprocity is a mechanism that can promote cooperation among populations in which individuals cannot repeatedly interact. Indirect reciprocity evaluates each individual's behaviour through social norms, based on which the reputation for each individual can be labelled. In the traditional models, it is usually assumed that all individual reputations are observable and are common knowledge to everyone in the population. In this paper, we relax this assumption and discuss an indirect-reciprocity model under incomplete information in which individuals have private opinions of others. Moreover, based on the observation that some people may have reservations about the current behaviour of an actor, which does not change their previous impressions on him, we generalize this phenomenon as withhold-judgment. We introduce punishment strategy and nine second-order social norms including withhold-judgment and explore how cooperation evolves in both public and private reputation scenarios. We find that social norms that allow for withhold-judgment can maintain high levels of cooperation. Although in the private reputation scenario, there is a situation in which more and more individuals have divarication over time, causing the reputation system to collapse, social norms that allow for withhold-judgment are still robust even if there are noise, variation, and incomplete information. In addition, we find that the introduction of punishment can promote cooperation, but in some situations, punishment will have a negative impact on social welfare.