This work presents an optimal control problem for an optional public goods game when a fractional punishment is used to improve cooperation. The objective function is mainly determined by the error trajectory, as well as the effort of the controller over a prespecified period. The results show that having chosen the parameters of the objective function appropriately, several optimal solutions can be obtained. Still, most importantly, given the desired kind of solution trajectory and the cost of investment in time, money, and capable personnel, the optimal control provides a more affordable solution (in terms of control effort) than the fractional punishment itself.

Punishment is probably the most frequently used mechanism to increase cooperation in Public Goods Games (PGG); however, it is expensive. To address this problem, this paper introduces an optimal control problem that uses fractional punishment to promote cooperation. We present a series of computational experiments illustrating the effects of single and combined terms of the optimization cost function. In the findings, the optimal controller outperforms the use of constant fractional punishment and gives an insight into the period and size of the penalization to be implemented with respect to the defection in the game.