  Combined prosocial incentives, integrating reward for cooperators and
punishment for defectors, are effective tools to promote cooperation among
competing agents in population games. Existing research concentrated on how to
adjust reward or punishment, as two mutually exclusive tools, during the
evolutionary process to achieve the desired proportion of cooperators in the
population, and less attention has been given to exploring a combined
incentive-based control policy that can steer the system to the full
cooperation state at the lowest cost. In this work we propose a combined
incentive scheme in a population of agents whose conflicting interactions are
described by the prisoner's dilemma game on complete graphs and regular
networks, respectively. By devising an index function for quantifying the
implementation cost of the combined incentives, we analytically construct the
optimally combined incentive protocol by using optimal control theory. By means
of theoretical analysis, we identify the mathematical conditions, under which
the optimally combined incentive scheme requires the minimal amount of cost. In
addition to numerical calculations, we further perform computer simulations to
verify our theoretical results and explore their robustness on different types
of network structures.
