Figure 5.

Evolution of cooperation under the influence of external incentives. Light solid lines indicate simulations whereas dash dot lines are theoretical results. During the evolution, we separately implement punishment (the fine is 0.4) and reward (the bonus is 0.65), or jointly enforce them in a population where the IPGG, dSH, PGG and dSD are played in each state with probability distribution (0.05, 0.05, 0.85, 0.05), respectively. The population structure is a lattice network. Parameter values: N = 400, d = 5, β = 0.05, $\mathcal{C}=c=1$ for all games, except r_s = 3 in the PGG and TPGG, r_s = 5 and T = [d/2] + 2 in the dSH, and ${\mathcal{B}}_s=12$ in the dSD.