Cooperation in the two-population snowdrift game with punishment enforced through different mechanisms

I study two mechanisms based on punishment to promote cooperation in the two-population snowdrift game. The first mechanism follows the traditional approach in the literature and is based on the inclusión of a third strategy in the payoff matrix of the stage-game. The second mechanism consists of letting cooperators to punish defectors with a given exogenous frequency. While both mechanisms share the same result regarding the minimum required level of punishment in order to avoid invasion of both populations by defectors, stability in the mechanism following the non-traditional 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. Results were obtained analytically through non-linear ordinary differential equations and laso using an agent-based simulation and there was a good level of agreement between both with respect to the evolutionary pattern over time and the possible steady-states.