Decentralized incentives for general well-being in networked public goods game

Prosocial incentives are recognized as effective tools to promote public cooperation. But their usage is costly, hence we need to find an optimal policy that offers the best outcome for a minimal cost. This problem has attracted extensive attention of scholars from many fields, and it has been studied intensively in well-mixed populations. Yet, the system behavior in the case of group interactions where decentralized incentives are involved is rather unexplored. To fill this gap, we consider the possible consequences of decentralized rewarding and punishing incentives in a spatially arranged population where players play public goods game with their neighbors. Importantly, both time-independent and dynamically varying cases are checked. With the help of pair approximation approach, we derive the dynamical equations under weak selection and obtain theoretical conditions of the minimally requested amounts for the expected outcome. By using optimal control theory, we also devise an index function to quantify the executing costs and obtain the optimal dynamical punishing and rewarding protocols. In addition, both numerical calculations and Monte Carlo simulations support that the optimal dynamical rewarding scheme requires a lower cumulative cost than the punishing alternative if the initial cooperation level is below a threshold value. Otherwise, the usage of punishment is more efficient