The impact of heterogeneous scale return coefficient between groups on the emergence of cooperation in spatial public goods game

The spatial public goods game (SPGG) is considered as a basic model for understanding the emergence of cooperation under the network reciprocity mechanism. For an intra-group game in SPGG, it is generally assumed that the total investment of all investors is linearly amplified and distributed within the group. In real situations, the value of public goods may be manifested as a nonlinear relationship of the total investment. In addition, the nonlinear function relationships are also heterogeneous for different public goods. Inspired by this observation, we made two improvements in the traditional SPGG. First, we introduced a scale return coefficient in the group of PGG to describe the nonlinear relationship between the value of public goods and the amount of investment. Then, we considered the heterogeneity of the scale return coefficient between different groups in SPGG. We investigated three types of distributions including uniform distribution, exponential distribution, and power-law distribution. Besides, we introduced an amplitude parameter to measure the degree of heterogeneity of the scale return coefficient. Simulation experiments on the square lattice network demonstrate that the larger the amplitude parameter for the same kind of distribution, which means that the stronger the heterogeneity, the more conducive to the cooperation of the population. Under the same amplitude, for different distributions, uniform distribution is most conducive to the evolution of cooperation in most parameter ranges. However, for some special parameter combinations, the exponential distribution may be the most beneficial to the emergence of cooperation. These results can be helpful to understand the impact of inter-group heterogeneity on the evolution of cooperation in SPGG.