Optimal Control of Fractional Punishment in Optional Public Goods Game

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.