Advanced defensive cooperators promote cooperation in the prisoner’s dilemma game

Abstract

There are four strategies in the population, including primary defensive cooperation, advanced defensive cooperation, anti-defensive defection and common defection. The primary defensive cooperators take defensive measures when they interact with the common defectors. To pursue higher payoffs, anti-defensive defectors disguise identity to get rid of the defense of the primary defensive cooperators. Nevertheless, they cannot escape from the defense of the advanced defensive cooperators, and will be punished. By undertaking a screening cost, the advanced defensive cooperators play a defensive role against the anti-defensive defectors and common defectors simultaneously. The effect of defense is reflected in the increase of the cooperator’s benefit and the decrease of the defector’s benefit. We use replication dynamics to study the evolution process of the strategies in the well-mixed population. It is found that the advanced defensive cooperators can effectively promote cooperation. The common defectors disappear at low temptation values. However, the fraction of common defectors is non-monotonic about defensive benefit at high temptation values. Interestingly, when the defensive loss is small, the increase in the fraction of anti-defensive defectors stimulates the prosperity of advanced defensive cooperators, which in turn inhibits the evolution of anti-defensive defectors. This process is called negative feedback. Primary defensive cooperators are effectively free-riding on advanced defensive cooperators under larger defensive loss and fine. In addition, reducing the screening cost can improve the cooperation level.