Evolutionary dynamics of spatial public goods games with insurance mechanisms

Insurance, a pivotal risk management tool in economic activities, protects participants in the public goods game from exploitation by defectors. In this study, we incorporate an insurance mechanism into spatial public goods games, empowering all participants to opt for insurance. By exploring phase transitions and spatial dynamics across various systems, we investigate the competition among multiple strategies within the insurance framework, revealing how insurance influences the evolution of cooperation. Our findings show that insured defectors can promote cooperation by creating a self-organized cyclic state. In contrast, under high synergy coefficients, insured cooperators may hinder the evolution of cooperation through indirect territorial competition. To account for insurers' profit motives, we further develop a more realistic two-layer network comprising both individual and insurance layers. This setup facilitates the examination of the coevolution of strategies between PGG participants and insurers on cooperation. Notably, this two-layer system exhibits more complex phase transitions, characterized by unique invasion dynamics that, under specific conditions, can further promote cooperation through insurers' strategy update.