Evolution of cooperation in spatial iterated Prisoner's Dilemma games under localized extremal dynamics

The spatial Iterated Prisoner's Dilemma game has been widely studied in order to explain the evolution of cooperation. Considering the large strategy space size and infinite interaction times, it is unrealistic to adopt the common imitate-best updating rule, which assumes that the human players have much stronger abilities to recognize their neighbors' strategies than they do in the one-shot game. In this paper, a novel localized extremal dynamic system is proposed, in which each player only needs to recognize the payoff of his neighbors and changes his strategy randomly when he receives the lowest payoff in his neighborhood. The evolution of cooperation is here explored under this updating rule for neighborhoods of different sizes, which are characterized by their corresponding radiuses  r. The results show that when r=1, the system is trapped in a checkerboard-like state, where half of the players consistently use AllD-like strategies and the other half constantly change their strategies. When r=2, the system first enters an AllD-like state, from which it escapes, and finally evolves to a TFT-like state. When r is larger, the system locks in a situation with similar low average fitness as r=1. The number of active players and the ability to form clusters jointly distinguish the evolutionary processes for different values of r from each other. The current findings further provide some insight into the evolution of cooperation and collective behavior in biological and social systems.