Effect of depreciation of the public goods in spatial public goods games

In this work, depreciated effect of the public goods is considered in the public goods games, which is realized by rescaling the multiplication factor r of each group as r' = r(nc/G)^beta (beat>= 0). It is assumed that each individual enjoys the full profit of the public goods if all the players of this group are cooperators, otherwise, the value of the public goods is reduced to r'. It is found that compared with the original version (beta = 0), emergence of cooperation is remarkably promoted for beta > 0, and there exit optimal values of beta inducing the best cooperation. Moreover, the optimal plat of beta broadens as r increases. Furthermore, effect of noise on the evolution of cooperation is studied, it is presented that variation of cooperator density with the noise is dependent of the value of beta and r, and cooperation dominates over most of the range of noise at an intermediate value of beta = 1.0. We study the initial distribution of the multiplication factor at beta = 1.0, and find that all the distributions can be described as Gauss distribution