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

Angsheng Li,Xi Yong,Gui-Quan Sun

doi:10.1371/journal.pone.0116429

Angsheng Li, Xi Yong + Show 1 more

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https://doi.org/10.1371/journal.pone.0116429

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Abstract

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 GPDW, GPDF and GPDN 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 γ 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.

Highlights

The prisoner’s dilemma (PD, for short) game is one of the well-known games, having implications in a wide range of disciplines
We found that our games have a new Nash equilibrium principle, and that entanglement is the principle of emergence and convergence of super cooperation
Our experiments demonstrate that for every temptation b, and for appropriately large entanglement γ, either super cooperation emerges or cooperation and super cooperation coexist and dominate the network in the evolutionary GPDW games, that for every benefit-cost ratio r in the normalised prisoner’s dilemma game, there is a phase transition for the entanglement γ at some threshold γ0 for evolutionary GPDN games, that for the evolutionary GPDW games with normalised payoff updating strategy, there is an interval (γ0, γ1) such that if the entanglement γ γ0, defection dominates the network, and if γ ! γ1, super cooperation dominates the network, and that the normalised payoff updating strategy plays no role in the evolutionary GPDN games on the scale-free networks

Summary

Introduction

The prisoner’s dilemma (PD, for short) game is one of the well-known games, having implications in a wide range of disciplines. In a PD game, two players simultaneously decide their strategy C (cooperator) or D (defector). For mutual cooperation, both players receive a reward R and receive punishment P upon mutual defection. If one cooperates and the other defects, the cooperator gains the lowest payoff S and the traitor gains temptation T. The normalized version of the prisoner’s dilemma game is usually defined by a parameter r, in which case, the reward R = 1, the temptation T = 1 + r, the punishment P = 0 and the lowest sucker’s payoff S = −r

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