The effect of increasing returns to scale in public goods investment on threshold values of cooperation under social exclusion mechanism

This paper establishes a stochastic evolutionary public goods game (PGG) model with exclusion-type strategies in a well-mixed and finite size population. The effect of increasing-returns-to-scale in public goods investment is considered. We focus on the stochastic stable equilibrium (SSE) of the evolutionary system. By numerical experiments, we observe that there are two types of SSE in the system, namely, “All individuals choosing defection” (“All D” state) and “the coexistence of cooperation and exclusion” (“C+E” states); and three types of phases, namely, choosing the “All D” state with probability one (“D” phase), choosing the “C+E” states with probability one (“C+E” phase) and choosing the “All D” and “C+E” states with non-zero probabilities (“C+E+D” phase). We study the combined effects of four parameters (probability of exclusion success, unit exclusion cost, increasing-returns-to-scale coefficient, investment amplification factor) on the phase selection of the system. We get the boundary curves or surfaces for each pair or set of combined parameters, thus conditions for the evolution of cooperative strategies depending on these parameters can be given. Notably, we observe that in a well-mixed population with exclusion-type strategies, the increasing-returns-to-scale coefficient cannot change the parameter boundary between the “C+E” phase and the “C+E+D” phase within our parameters. Corresponding simulation experiments are also carried out in a structured population, and it is found that the increasing-returns-to-scale effect can greatly reduce critical values of the amplification factor under which cooperation can emerge, and the increase in the increasing-returns-to-scale coefficient can induce the system to transfer directly from the “D” phase to the “C+E” phase, which is different from the well-mixed situation. This research can help us better understand the emergence of cooperation in the PGG with increasing-returns-to-scale effect under social exclusion mechanism.