Free-rider or contributor: A dilemma in spatial threshold public goods games

Late movers refer to individuals who adjust their strategies after observing the choices of others in a public goods game, can significantly impact the likelihood of public goods provision. This creates a social dilemma: late movers act as free-riders to rely on others to contribute, yet they also risk the public good not being provided if contributions fall short. To address this dilemma, we utilize a spatial threshold public goods game model to examine how the behaviors of late movers influence overall cooperative levels within a population. We propose a probabilistic rule for late movers that captures the dilemma between cooperating to enhance the collective good and free-riding to maximize individual benefits. Our findings indicate that strategic late movers can significantly enhance cooperation by adapting to local cooperation levels. However, an excessively high proportion of late movers may reduce the overall level of cooperation. Additionally, the formation of clusters by late movers results in higher cooperative levels compared to random distribution. Our study provides novel insights into the dynamics of cooperation within the framework of spatial threshold public goods games. By incorporating the role of late movers, we highlight how adaptive strategies can mitigate the free-rider problem and promote cooperative behaviors.