Abstract
The authors proposed a quantum Prisoner's Dilemma (PD) game as a natural extension of the classic PD game to resolve the dilemma. Here, we establish a new Nash equilibrium principle of the game, propose the notion of convergence and discover the convergence and phase-transition phenomena of the evolutionary games on networks. We investigate the many-body extension of the game or evolutionary games in networks. For homogeneous networks, we show that entanglement guarantees a quick convergence of super cooperation, that there is a phase transition from the convergence of defection to the convergence of super cooperation, and that the threshold for the phase transitions is principally determined by the Nash equilibrium principle of the game, with an accompanying perturbation by the variations of structures of networks. For heterogeneous networks, we show that the equilibrium frequencies of super-cooperators are divergent, that entanglement guarantees emergence of super-cooperation and that there is a phase transition of the emergence with the threshold determined by the Nash equilibrium principle, accompanied by a perturbation by the variations of structures of networks. Our results explore systematically, for the first time, the dynamics, morphogenesis and convergence of evolutionary games in interacting and competing systems.
Highlights
The Prisoner’s Dilemma (PD) game is one of the well-known games, having implications in a wide range of disciplines
It was shown that both versions of the quantum PD games have a new Nash equilibrium principle, and that for heterogeneous networks of the preferential attachment (PA) model [7], and for appropriately large entanglement, supercooperation quickly emerges in evolutionary quantum PD games on the networks
We propose the notion of convergence of evolutionary games on networks, and investigate the convergence of evolutionary quantum PD games on networks of the classical models
Summary
The Prisoner’s Dilemma (PD) game is one of the well-known games, having implications in a wide range of disciplines. Different updating strategies may play a role in the emergence of cooperation in evolutionary games on networks [18] In all of these studies, the PD game is the weak version defined by Nowak & May [1]. It was shown that both versions of the quantum PD games have a new Nash equilibrium principle, and that for heterogeneous networks of the preferential attachment (PA) model [7], and for appropriately large entanglement, supercooperation quickly emerges in evolutionary quantum PD games on the networks. This is the first time that evolutionary games on networks have been studied for the normalized full PD games. Our results establish a new theory of the convergence and phase transition of evolutionary games in networks
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