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.

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In a group of individuals that come together to produce a good or provide a service, the cooperators (who pay to produce the good) are often exploited by those who receive the benefit without paying the cost. Models were developed over time using incentives (reward or punishment) and the option of abandoning the initiative to promote and stabilize the cooperation. In this paper we analyze several models based on the evolutionary game theory and public good games. We compare and organize them in a taxonomic table following their main characteristics to select the most suitable for a specific problem. The analyzed models are compared by using a public good problem in community projects for water supply. We have reasonable assurance that phenomena that appear on models also occurs in these community projects. Therefore, we propose that evolutionary game theory can be a useful tool for policy-makers in order to improve cooperation and discourage defection in sanitation boards.