Exploring optimal institutional incentives for public cooperation

Prosocial incentive can promote cooperation, but providing incentive is costly. Institutions in human society may prefer to use an incentive strategy which is able to promote cooperation at a reasonable cost. However, thus far few works have explored the optimal institutional incentives which minimize related cost for the benefit of public cooperation. In this work, in combination with optimal control theory we thus formulate two optimal control problems to explore the optimal incentive strategies for institutional reward and punishment respectively. By using the approach of Hamilton-Jacobi-Bellman equation for well-mixed populations, we theoretically obtain the optimal positive and negative incentive strategies with the minimal cumulative cost respectively. Additionally, we provide numerical examples to verify that the obtained optimal incentives allow the dynamical system to reach the desired destination at the lowest cumulative cost in comparison with other given incentive strategies. Furthermore, we find that the optimal punishing strategy is a cheaper way for obtaining an expected cooperation level when it is compared with the optimal rewarding strategy.