Exhaustive classification of stochastic cooperative-level dependent strategies in iterated prisoner’s dilemma

Abstract

We propose an infinitely iterated prisoner’s dilemma in which the stochastic strategies react to a global cooperative level. The cooperative level is a result of the collective actions of the agents, not only of the opponents. The stochastic strategies are represented by $(y,p,q)$, where $p$ ($q$) is the probability of taking a cooperative action when the cooperative level is below (equal or above) a threshold $m_c$, and $y$ is an initial cooperative probability. A relevant situation, among others, is that of budding yeast growing on sucrose, where the monosaccharide created by some yeasts are shared by those in the neighborhood. The decision on whether to create (to cooperate) monosaccharide or not (to defect) is dependent on the monosaccharide concentration and thus can be modeled by stochastic cooperative-level dependent strategies. We classify such stochastic strategies exhaustively for different values of $m_c$. All Nash equilibria are identified and strategies that are stable against invasion by selection pressure are characterized. The results are helpful in understanding the final winning strategies emerged in competing systems in which actions are decided based on the cooperative level.