Cooperative behavior in evolutionary snowdrift games with the unconditional imitation rule on regular lattices

We study an evolutionary snowdrift game with the unconditional imitation updating rule on regular lattices. Detailed numerical simulations establish the structure of plateaus and discontinuous jumps of the equilibrium cooperation frequency ${f}_{c}$ as a function of the cost-to-benefit ratio $r$. By analyzing the stability of local configurations, it is found that the transitions occur at values of $r$ at which there are changes in the ranking of the payoffs to the different local configurations of agents using different strategies. Nonmonotonic behavior of ${f}_{c}(r)$ at the intermediate range of $r$ is analyzed in terms of the formation of blocks of agents using the cooperative strategy that are stabilized by agents inside the block due to the updating rule. For random initial condition with 50%--50% agents of different strategies randomly dispersed, cooperation persists in the whole range of $r$ and the level of cooperation is higher than that in the well-mixed case in a wide range of $r$. These results are in sharp contrast to those based on the replicator updating rule. The sensitivity to initial states with different fractions of cooperative agents is also discussed. The results serve to illustrate that both the spatial structure and the updating rule are important in determining the level of cooperation in a competing population. When extreme initial states are used where there are very few agents of a strategy in a background of the opposite strategy, the result would depend on the stability of the clusters formed by the initially minority agents.