Entanglement plays an important role in evolutionary generalized prisoner's dilemma game on small-world networks

Cooperation is a fundamental aspect of biological systems and human society. In this work, we propose a generalized prisoner's dilemma quantum game and study the game on Watts-Strogatz small-world network (WS model) with different values of entanglement. Here entanglement is used as a metaphor to study relationship between players. In a single step, each player has three strategies to choose: cooperation, defection and adopting a special quantum strategy first introduced by Eisert (denoted by Q). The person who chooses the quantum strategy Q can be regarded as a super cooperator. In prisoner's dilemma game, cooperation is very hard to emerge on relatively homogeneous networks, such as random graphs and WS model. Our experimental results show that in generalized prisoners dilemma quantum game, entanglement can guarantee emergence of super cooperators and greatly increase the payoff of the whole network. Even on homogeneous networks with very large inducement value(b is close to 2), the frequency of cooperators can still exceeds 70%. This work provides a new insight into understanding cooperation when considering the role of relationship between cooperators.