Aspiration Can Promote Cooperation in Well-Mixed Populations As in Regular Graphs

Classical studies on aspiration-based dynamics suggest that dissatisfied individuals switch their strategies without taking into account the success of others. The imitation-based dynamics allow individuals to imitate successful strategies without taking into account their own-satisfactions. In this article, we propose to study a dynamic based on aspiration, which takes into account imitation of successful strategies for dissatisfied individuals. Individuals compare their success to their aspired levels. This mechanism helps individuals with a minimum of self-satisfaction to maintain their strategies. Dissatisfied individuals will learn from their neighbors by choosing the successful strategies. We derive an exact expression of the fixation probability in well-mixed populations as in graph-structured populations. As a result, we show that weak selection favors the evolution of cooperation if the difference in aspired level exceeds some crucial value. Increasing the aspired level of cooperation should oppose cooperative behavior while increasing the aspired level of defection should promote cooperative behavior. We show that the cooperation level decreases as the connectivity increases. The best scenario for the cooperative evolution is a graph with a small connectivity, while the worst scenario is a well-mixed population.