In Public Goods Games (PGG), the temptation to free-ride on others’ contributions poses a significant threat to the sustainability of cooperative societies. Therefore, societies strive to mitigate this through incentive systems, employing rewards and punishments to foster cooperative behavior. Thus, peer punishment, in which cooperators sanction defectors, as well as pool punishment, where a centralized punishment institution executes the punishment, is deeply analyzed in previous works. Although the literature indicates that these methods may enhance cooperation on social dilemmas under particular contexts, there are still open questions, for instance, the structural connection between graduated punishment and the monitoring of public goods games. Our investigation proposes a compulsory PGG framework under Panoptical surveillance. Inspired by Foucault’s theories on disciplinary mechanisms and biopower, we present a novel mathematical model that scrutinizes the balance between the severity and scope of punishment to catalyze cooperative behavior. By integrating perspectives from evolutionary game theory and Foucault’s theories of power and discipline, this research uncovers the theoretical foundations of mathematical frameworks involved in punishment and discipline structures. We show that well-calibrated punishment and discipline schemes, leveraging the panoptical effect for universal oversight, can effectively mitigate the free-rider dilemma, fostering enhanced cooperation. This interdisciplinary approach not only elucidates the dynamics of cooperation in societal constructs but also underscores the importance of integrating diverse methodologies to address the complexities of fostering cooperative evolution.

Improving and maintaining cooperation are fundamental issues for any project to be time-persistent, and sanctioning free riders may be the most applied method to achieve it. However, the application of sanctions differs from one group (project or institution) to another. We propose an optional, public good game model where a randomly selected set of the free riders is punished. To this end, we introduce a parameter that establishes the portion of free riders sanctioned with the purpose to control the population state evolution in the game. This parameter modifies the phase portrait of the system, and we show that, when the parameter surpasses a threshold, the full cooperation equilibrium point becomes a stable global attractor. Hence, we demonstrate that the fractional approach improves cooperation while reducing the sanctioning cost.

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.

From the harmonies that govern the behavior of an ant colony to the synchronized flight of a fleet of birds, nature eloquently demonstrates the transformative power of cooperation. It is not merely an idea but a tangible force that drives the pulse of lifeguides survival, and orchestrates the mighty symphony of evolution. Cooperation is, therefore, not a choice but a requirement— a fundamental tenet of our existence — underpinning the evolution of societies and the prevailing of nature over vast odds. Far from mere instruments of coercion, control, and punishment mechanisms are crucial safeguards in our society, nurturing cooperation and upholding justice. They discourage individuals from exploiting social frameworks and sustaining cooperation norms. In the context of a surveillance strategy designed to foster cooperation, a panoptical approach refers to a system in which control is always possible. This method creates a state of constant and deliberate visibility, which ensures that power operates automatically. It minimizes the punishment costs by making the ubiquitous impression of being watched and held accountable, encouraging cooperative conduct. In a previous paper, we presented a punishment mechanism under panoptic surveillance in which the number of penalized individuals is reduced to an adjustable fraction of the population of free riders from which the sanctioned are randomly selected. The findings imply that by adjusting "how many" free riders are punished, potential sanctions alone can increase population-wide cooperation. This article builds on that strategy by examining the influence of "how many" free riders are punished and analyzes the impactof "how much" the punitive sanction is severe on encouraging cooperation. We employ a compulsory public goods game model with replicator dynamics to investigate this issue. The findings suggest that long-term sustainable cooperation can be achievedby fine-tuning the compromise between retaliation severity and punishment certainty (the number of sanctioned wrongdoers),i.e., the delicate nuance of mercy and justice.

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.