Global Convergence for Replicator Dynamics of Repeated Snowdrift Games

To understand the emergence and sustainment of cooperative behavior in interacting collectives, we perform global convergence analysis for replicator dynamics of a large, well-mixed population of individuals playing a repeated snowdrift game with four typical strategies, which are always cooperate (ALLC), tit-for-tat (TFT), suspicious tit-for-tat (STFT), and always defect (ALLD). The dynamical model is a 3-D ordinary differential equation (ODE) system that is parameterized by the payoffs of the base game. Instead of routine searches for evolutionarily stable strategies and sets, we expand our analysis to determining the asymptotic behavior of solution trajectories starting from any initial state, and in particular, show that for the full range of payoffs, every trajectory of the system converges to an equilibrium point. What enables us to achieve such comprehensive results is studying the dynamics of two ratios of the state variables, each of which either monotonically increases or decreases in the half-spaces separated by their corresponding planes. The convergence results highlight two findings. First, the inclusion of TFT- and STFT-players, the two types of conditional strategy players in the game, increases the share of cooperators of the overall population compared to the situation when the population consists of only ALLC and ALLD-players. Second, surprisingly enough, regardless of the payoffs, there always exists a set of initial conditions under which ALLC-players do not vanish in the long run, which does not hold for any of the other three types of players.