Costly punishment and cooperation in the evolutionary snowdrift game

The role of punishments in promoting cooperation is an important issue. We incorporate costly punishments into the snowdrift game (SG) by introducing a third punishing (P) character, and study the effects. The punishers, who carry basically a cooperative (C) character, are willing to pay a cost α so as to punish a non-cooperative (D) opponent by β. Depending on the initial fractions of the characters, α, β, and the cost-to-benefit ratio r in the SG, the three-character system evolves into a steady state consisting either only of C and P characters or only of C and D characters, in a well-mixed population. The former situation represents an enhancement in cooperation relative to the SG, while the latter is similar to the SG. The dynamics in approaching these different steady states are found to be different. Analytically, the key features in the dynamics and the steady states observed in simulations are captured by a set of differential equations. The sensitivity to the initial distribution of characters is studied by depicting the flow in a phase portrait and analyzing the nature of fixed points. The analysis also shows the role of P-character agents in preventing a system from invasion by D-character agents. Starting from a population consisting only of C and P agents, a D-character agent intended to invade the system cannot survive when the initial fraction of P agents is greater than r/β. Our model, defined intentionally as a simulation algorithm, can be readily generalized to incorporate many interesting effects, such as those in a networked population.