While the free-rider problem has been studied extensively ([Botta 2013, Yamagishi 1986, Botta 2021]), this has also increased the number of theories and questions to be answered around it. One way in which the free-rider problem is modeled is through a Public Goods Game, in which contributions made to a common pool of resources by players are then scaled and distributed equally to all participants.

In this work, we focus on the "optional" variant of the Public Goods Game, where players may opt for a risk-free (yet often lower) reward instead of participating in the game. Building on [Botta 2024] and [Grau 2022], we combine fractional (targeting a subset of free-riders) and incomplete (reducing benefits while imposing fines) punishment via optimal control to enhance cooperation in the entire population. Our mechanism mirrors real-world practices and helps to save the financial amount spent on penalizing just a part of the total number of free riders while
still obtaining the desired cooperation.

To tune the combined policy, we design an appropriate objective function to solve a problem of optimization with restriction, where the restriction is given by the replicator dynamics problem, including the fractional and incomplete punishment.

The findings provide policymakers with actionable insights; while stronger instantaneous punishments increase cooperation (and reduce the overall costs of punishment in the long term), they imply higher initial investments. Even though high cost can be challenging for managers, especially when free-riders make up a large part of the group, the proposed control mechanism offers a cost-effective alternative by strategically targeting only a fraction of free-riders or adjusting the fines, balancing enforcement costs with long-term cooperative outcomes.