Fractional punishment of free riders to improve cooperation in optional public good games

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.