Patch dynamics based on Prisonerʹs Dilemma game: superiority of golden rule

There has been much literature on ecological model of Prisonerʹs Dilemma (PD) game. This game illustrates that cooperation can evolve in situations where individuals tend to look after themselves. In order to explain some behaviors of altruism in animal societies, the strategy All Cooperate (AC), often called the Golden Rule, is more appropriate than other strategies. However, very little is known about the superiority of AC. In the present article, we study patch dynamics based on non-iterated PD game, applying two different methods: island and lattice models. Each patch is assumed to be either vacant or composed of a population of AC or All Defect (AD), where AD means a selfish strategy. Both models exhibit a phase transition between a phase where both AC and AD survive, and a phase where AD is extinct. The latter phase means that AC beats AD completely. In the case of lattice model, the extinction of AD easily occurs and the abundance of AC takes a larger value, compared with the island model. Our models can be also extended to general iterated PD game; we describe the reason why AC can outperform any other strategy.