Evolutionary dynamics in the spatial public goods game with tolerance-based expulsion and cooperation

Altruistic sanctions for free riders are among the most valuable approaches for promoting cooperation in human society. As a new form of punishment, expulsion allows individuals to expel defectors from their neighborhoods, which is considered an effective mechanism to promote the evolution of cooperation. Inspired by the fact that people often evaluate their surrounding environment before making decisions, we introduce a tolerance-based expulsion mechanism into the spatial public goods game. The condition for expulsion to be triggered is the number of defectors in the group exceeding the tolerance threshold. Unlike the traditional expulsion mechanism that requires the immediate removal of free rides from the group, tolerance provides a way for players to switch between pure cooperation and expulsion. Apparently, tolerance reduces the frequency of costly expulsions, providing a way for expellers to balance payoff and expulsion costs. Our simulations successfully reveal that tolerance-based expulsion can significantly foster cooperation and stabilize pure cooperation under negative conditions. Moreover, the optimal threshold for implementing expulsion is determined. By drawing the formation process of an effective resistance alliance, we prove that unselfish expellers can assist other pure cooperators in removing defection and using vacant sites to form an isolated layer. Interestingly, a pure cooperation cluster might provide a special shield for defection that result in poor expulsion performance. We also observe in the phase diagrams the continuous and discontinuous phase transitions between frozen and dynamical stationary states of the system. Our work extends the form of expulsion to a realistic perspective, and the results might help enhance the understanding of the expulsion mechanism.