Multiple tolerances dilute the second order cooperative dilemma

A peer punisher directly imposes fines upon defectors at a cost to himself. It is one of the mechanisms promoting cooperation, which is ubiquitous in nature. Typically, it is assumed that a peer punisher punishes provided that there is one defector in the group. The threshold that triggers punishment, however, is not necessarily one. The larger the threshold is, the more tolerant the peer punisher is. We study the evolutionary dynamics of those diverse tolerant peer punishment strategies in public goods game. We find that, i) less tolerant punishers prevail over tolerant ones; ii) large group size could enhance punishment, in contrast with the case in the first-order cooperative dilemma. Our analytical results are based on weak selection limit and large population size, which are verified by simulations. Our work sheds light on how punishment of diverse tolerance evolves.