Emergence of super cooperation of prisoner's dilemma games on scale-free networks

Recently, the authors proposed a quantum prisoner's dilemma game based on the spatial game of Nowak and May, and showed that the game can be played classically. By using this idea, we proposed three generalized prisoner's dilemma (GPD, for short) games based on the weak Prisoner's dilemma game, the full prisoner's dilemma game and the normalized Prisoner 's dilemma game, written by GPD_W, GPD_FandGPD_N respectively. Our games consist of two players, each of which has three strategies: cooperator (C), defector (D) and super cooperator (denoted by Q), and have a parameter ã to measure the entangled relationship between the two players. We found that our generalised prisoner 's dilemma games have new Nash equilibrium principles, that entanglement is the principle of emergence and convergence (i.e., guaranteed emergence) of super cooperation in evolutions of our generalised prisoner 's dilemma games on scale-free networks, that entanglement provides a threshold for a phase transition of super cooperation in evolutions of our generalised prisoner's dilemma games on scale-free networks, that the role of heterogeneity of the scale-free networks in cooperations and super cooperations is very limited, and that well-defined structures of scale-free networks allow coexistence of cooperators and super cooperators in the evolutions of the weak version of our generalised prisoner's dilemma games.