Depreciation of public goods in spatial public goods games

In real situations, the value of public goods will be reduced or even lost because of external factors or for intrinsic reasons. In this work, we investigate the evolution of cooperation by considering the effect of depreciation of public goods in spatial public goods games on a square lattice. It is assumed that each individual gains full advantage if the number of the cooperators n(c) within a group centered on that individual equals or exceeds the critical mass (CM). Otherwise, there is depreciation of the public goods, which is realized by rescaling the multiplication factor r to (n(c)/CM)r. It is shown that the emergence of cooperation is remarkably promoted for CM > 1 even at small values of r, and a global cooperative level is achieved at an intermediate value of CM = 4 at a small r. We further study the effect of depreciation of public goods on different topologies of a regular lattice, and find that the system always reaches global cooperation at a moderate value of CM = G - 1 regardless of whether or not there exist overlapping triangle structures on the regular lattice, where G is the group size of the associated regular lattice.