Evolutionary dynamics in state-feedback public goods games with peer punishment.

Public goods game serves as a valuable paradigm for studying the challenges of collective cooperation in human and natural societies. Peer punishment is often considered an effective incentive for promoting cooperation in such contexts. However, previous related studies have mostly ignored the positive feedback effect of collective contributions on individual payoffs. In this work, we explore global and local state-feedback, where the multiplication factor is positively correlated with the frequency of contributors in the entire population or within the game group, respectively. By using replicator dynamics in an infinite well-mixed population, we reveal that state-based feedback plays a crucial role in alleviating the cooperative dilemma by enhancing and sustaining cooperation compared to the feedback-free case. Moreover, when the feedback strength is sufficiently strong or the baseline multiplication factor is sufficiently high, the system with local state-feedback provides full cooperation, hence supporting the "think globally, act locally" principle. Besides, we show that the second-order free-rider problem can be partially mitigated under certain conditions when the state-feedback is employed. Importantly, these results remain robust with respect to variations in punishment cost and fine.