Particle swarm intelligence and the evolution of cooperation in the spatial public goods game with punishment

Public goods game has been widely studied on the formation mechanism of cooperation in social dilemmas. Extensive experimental and theoretical researches have shown that punishment can significantly facilitate cooperation. In nature and society, the individual continuously adjusts the degree of cooperation and severity of punishment. In this paper, the particle swarm optimization (PSO) is used for describing the learning rule of updating the input of cooperation and punishment cost. Cooperators performing both cooperation and punishment make a trade-off between public goods investment and punishment cost. We examine how the PSO learning rule affects the evolution of cooperation and punishment intensity. Our simulation results show that the intermediate values of weighting coefficient omega increase the input of punishment. More cooperators in the interior of clusters input punishment cost, which decreases the cooperation level. For low or high values of omega, cooperators decrease the input of punishment and only cooperators on the edge of clusters incline to punish defectors, which increases the cooperation level of population. Our research provides new insights on the coexistence of cooperation and punishment. (C) 2021 Elsevier Inc. All rights reserved.