Cooperative behavior in NN-person evolutionary snowdrift games with punishment

• A punishing strategy is included into NN-person evolutionary snowdrift game.
• Cooperative, non-cooperative and punishing strategies cannot coexist.
• Frozen or dynamical phase may result depending on model parameters.
• Two phases with different sensitivities on initial conditions.
• Simulation algorithm proposed for model with validity confirmed.

We generalize the NN-person evolutionary snowdrift game to incorporate the effects of costly punishment in a well-mixed population. A set of dynamical equations that account for the evolution of the frequencies of the three strategies under replicator dynamics is formulated. At long time, the system evolves into one of two phases with different properties consisting only of two strategies, and three-strategy coexistence is not allowed. Small cost-to-benefit ratio, big competing group size, and severe punishment tend to suppress non-cooperators, and lead to a cooperative system with a mixture of cooperators and punishers. The resulting composition depends on the initial conditions as the dynamics is frozen once non-cooperators extinct. Large cost-to-benefit ratio, small competing group size, and light punishment tend to be self-destructive for the punishers, and lead to a mixture of cooperators and non-cooperators with composition independent of initial conditions and a continual dynamics. The frozen phase and dynamical phase correspond to a line of fixed points and a single fixed point on different axes in the phase space, respectively. A simulation algorithm that mimics the replicator dynamics exactly is proposed. Results of the dynamical equations and numerical simulations are found to be in exact agreement.