Cooperation In The Well-Mixed Two-Population Snowdrift Game With Punishment Enforced Through Different Mechanisms

I study two mechanisms based on punishment to promote cooperation in the well-mixed two-population snowdrift game (SG). The first mechanism follows a standard approach in the literature and is based on the inclusion of a third additional pure strategy in the payoff matrix of the stage-game. Differently, the second mechanism consists of letting cooperators punish defectors with a given exogenous frequency. In the latter, the pure strategy cooperation is replaced by a mixed strategy in which cooperators randomize between cooperation and punishment against defectors. While both mechanisms share the same result regarding the minimum required level of punishment in order to eliminate defectors in both populations, stability in the mechanism following the second approach is more robust in the sense that extinction of defectors is a globally asymptotically stable state for any interior initial conditions in the phase space. Thus, the second mechanism displays a topologically simpler model but the robustness of the evolutionary equilibrium is improved. Results were obtained analytically through nonlinear differential equations and also using an agent-based simulation. There was a good level of agreement between both approaches with respect to the evolutionary pattern over time and the possible steady-states.