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

In this work, the depreciation effect of public goods is considered in the public goods games, which is realized by rescaling the multiplication factor r of each group as r′=r(ncG)β (β≥0). It is assumed that each individual enjoys the full profit r of the public goods if all the players of this group are cooperators. Otherwise, the value of public goods is reduced to r′. It is found that compared with the original version (β=0), the emergence of cooperation is remarkably promoted for β>0, and there exist intermediate values of β inducing the best cooperation. Particularly, there exists a range of β inducing the highest cooperative level, and this range of β broadens as r increases. It is further presented that the variation of cooperator density with noise has close relations with the values of β and r, and cooperation at an intermediate value of β=1.0 is most tolerant to noise.