Spatial public goods game with continuous contributions based on Particle Swarm Optimization learning and the evolution of cooperation

How to explain the evolution of cooperation in social dilemmas has plagued the theorists of various disciplines. In this paper, we study the public goods game on two structured populations, assuming that each individual can make a continuous contribution in its strategy space. In our model, individuals have memory and imitation ability. They can remember their best history strategy and learn their best neighbors by obtaining the local information. We use the particle swarm optimization (PSO) algorithm to simulate the learning process of individuals. Two network structures of the square lattice and the nearest-neighbor coupled network are considered respectively. We investigate the combined effects of memory, the invest enhancement and the network structure on the average cooperation level of the population. By simulation experiments, we find that the PSO learning mechanism can promote the evolution of cooperation in a large range of parameters under both of the two kinds of networks. Compared with the square lattice network, the nearest-neighbor coupled network can promote the evolution of cooperation in a larger parameter scope. Compared with the Fermi rule and the Genetic Algorithm learning, the PSO learning can induce the population achieve a wider range of cooperation level, which makes it possible to achieve a higher level of cooperation. These results are conducive to a better understanding of the emergence of cooperation in the real world.