State-dependent optimal incentive allocation protocols for cooperation in public goods games on regular networks

Incentives are often used to promote cooperation in a population of competing agents. Furthermore, it is a common method to combine both punishment and reward. However, how to optimally allocate an amount of incentive budget as reward and punishment to enhance cooperation in structured populations is still a challenging task. To address this problem, we here consider evolutionary public goods games on regular networks and assume that a fixed budget of incentives is dynamically allocated to reward cooperators or to punish defectors in a group of players, depending on the actual cooperation level in the population. By means of the pair approximation approach, we derive the dynamical equation for depicting the evolutionary dynamics of cooperation. We then formulate two optimal incentive allocation problems by minimizing the payoff difference between defectors and cooperators and maximizing the gradient of selection, respectively. We theoretically derive the optimal incentive allocation protocols for the two objective functions we considered. We find that the obtained protocols both depend sensitively on the efficiency ratio of reward to punishment. In addition, we provide numerical calculations to verify our theoretical results.