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Combined prosocial incentives, integrating reward for cooperators andpunishment for defectors, are effective tools to promote cooperation amongcompeting agents in population games. Existing research concentrated on how toadjust reward or punishment, as two mutually exclusive tools, during theevolutionary process to achieve the desired proportion of cooperators in thepopulation, and less attention has been given to exploring a combinedincentive-based control policy that can steer the system to the fullcooperation state at the lowest cost. In this work we propose a combinedincentive scheme in a population of agents whose conflicting interactions aredescribed by the prisoner's dilemma game on complete graphs and regularnetworks, respectively. By devising an index function for quantifying theimplementation cost of the combined incentives, we analytically construct theoptimally combined incentive protocol by using optimal control theory. By meansof theoretical analysis, we identify the mathematical conditions, under whichthe optimally combined incentive scheme requires the minimal amount of cost. Inaddition to numerical calculations, we further perform computer simulations toverify our theoretical results and explore their robustness on different typesof network structures.

Optimally combined incentive for cooperation among interacting agents in population games