Abstract
Punishment is widely recognized as an effective approach for averting from exploitation by free-riders in human society. However, punishment is costly, and thus rational individuals are unwilling to take the punishing action, resulting in the second-order free-rider problem. Recent experimental study evidences that individuals prefer conditional punishment, and their punishing decision depends on other members’ punishing decisions. In this work, we thus propose a theoretical model for conditional punishment and investigate how such conditional punishment influences cooperation in the public goods game. Considering conditional punishers only take the punishing action when the number of unconditional punishers exceeds a threshold number, we demonstrate that such conditional punishment induces the effect of a double-edged sword on the evolution of cooperation both in well-mixed and structured populations. Specifically, when it is relatively easy for conditional punishers to engage in the punishment activity corresponding to a low threshold value, cooperation can be promoted in comparison with the case without conditional punishment. Whereas when it is relatively difficult for conditional punishers to engage in the punishment activity corresponding to a high threshold value, cooperation is inhibited in comparison with the case without conditional punishment. Moreover, we verify that such double-edged sword effect exists in a wide range of model parameters and can be still observed in other different punishment regimes.
Highlights
The solutions to many challenges in human societies, such as the management of public resources[1,2,3] and the global warming[4,5,6], all boil down to resort to a broad collective action of cooperation
We propose a theoretical model for conditional punishment in the context of public goods games, and consider that conditional punishers will participate in the punishment activity with other unconditional punishers only when the number of unconditional punishers in the group is not less than a threshold number, otherwise they will just cooperate
We find that when conditional punishment is considered, the system will evolve to either the state of all defectors or the coexistence state of cooperators and unconditional punishers, no matter whether the threshold value is low or high (Fig. 1(a) and (b))
Summary
The solutions to many challenges in human societies, such as the management of public resources[1,2,3] and the global warming[4,5,6], all boil down to resort to a broad collective action of cooperation. While general civilians may hesitate to engage in sanction and their punishing decisions to robbers should significantly depend on the number of individuals who perform the punishment This novel behavior among punishers is completely distinct from the coordinated punishment investigated in some aforementioned works[34,37,38]. It still remains unclear how such conditional punishment, under which whether to sanction free-riders or not depends on the number of unconditional punishers in the group, influences the evolution of cooperation. The sum of all contributions in the group will be multiplied by a synergy factor r, and allotted among all group members irrespective of their contributions
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