Cooperative Enhancement of Cost Threshold in the Spatial N-Person Snowdrift Game

We study the cooperative behaviour of the N-person snowdrift game on a square lattice with different neighbourhoods and the cost threshold M for the cooperators. We apply the imitation mechanism for the players and find that there is an optimal frequency of cooperation f_C as M varies in the game and the peak of f_C shifts to higher M when more neighbours on the lattice are involved in the game. For a given cost-to-benefit ratio r, the system shows discontinuous phase transitions and the behaviour of f_C vs M shows step-like structures. We construct payoff level structures and find that the above features can be understood by analyzing the transition behaviour between the payoff levels. The variations of r and M are equivalent to tune the payoff level structures. The plateaus in the step-like structure have similar payoff level structures and the discontinuous jumps correspond to different payoff level structures.