A key aspect of the human society is the extensive cooperation among the genetically unrelated individuals. It is suggested that punishment promotes cooperation even in anonymous one-shot games where neither direct nor indirect reciprocity is possible, because in the presence of punishers, cooperators can gain higher payoffs than defectors. However, recent studies suggest that the positive effects of punishment on cooperation were possibly exaggerated because previous literature overlooked the potential detrimental effects of punishment and the existence of destructive punishment.

The detrimental effects of punishment are related to the lack of guidance of social norms or the high cost of punishment. Previous studies emphasized the role of punishment in the maintenance of social norms, but recent empirical studies have found that social norms have a guiding role in the implementation of punishment. The absence of social norms will lead to punishment being perceived as unfair by individuals, which undermines cooperation among individuals. The high cost of punishment is another issue that has been addressed in recent studies, which suggest that the administration of punishment reduces the average payoff of the group even if it can increase the amount of cooperation, particularly, in those games with only a few iterations. Therefore, cooperation can only be maintained by punishment if the cost for the punisher is low and the impact on the punished is high.

In addition, most previous studies have unreasonably assumed that punishment is always targeted at defectors. The relaxation of this assumption leads to the appearance of the so-called destructive punishment, which is usually (but not always) carried out by non-cooperators and motivated by revenge, material gains or spite. Destructive punishment includes counter-punishment (punishment targeted at punishers by the punished), antisocial punishment (punishment targeted at cooperators), blind punishment (indiscriminate punishment by the punished) and spiteful punishment (punishment by defectors driven by strategic reasons). The findings of some important studies suggest that punishment does not promote cooperation anymore when the destructive punishment is possible. The role of the destructive punishment in the evolution of cooperation raises a challenging question concerning the relation among the punishment, destructive punishment and cooperation. Some researchers argue that second-order free-riding on destructive punishment can restore the effectiveness of punishment, whereas others maintain that destructive punishment can prevent the co-evolution of cooperation and punishment. More studies are needed to better understand the role of destructive punishment.

By combining the findings of different studies, this paper raises two important issues that should be addressed. Firstly, different terms such as punishment, peer punishment or costly punishment are used to refer to punishment that promotes cooperation among individuals. However, a more precise definition is urgently needed to draw a clear distinction between punishment and destructive punishment, because logically speaking, destructive punishment is also a type of punishment, peer punishment or costly punishment. Secondly, the two prevailing approaches in the research on punishment, controlled laboratory experiment and agent-based simulation, usually lead to contradictory conclusions. Therefore, how to reconcile the research results from different approaches is another question that needs to be solved in future research.