Evolutionary dynamics of cooperation on interdependent networks with Prisoner's Dilemma and Snowdrift Game

The world in which we are living is a huge network of networks and should be described by interdependent networks. The interdependence between networks significantly affects the evolutionary dynamics of cooperation on them. Meanwhile, due to the diversity and complexity of social and biological systems, players on different networks may not interact with each other by the same way, which should be described by multiple models in evolutionary game theory, such as the Prisoner’s Dilemma and Snowdrift Game. We therefore study the evolutionary dynamics of cooperation on two interdependent networks playing different games respectively. We clearly evidence that, with the increment of network interdependence, the evolution of cooperation is dramatically promoted on the network playing Prisoner’s Dilemma. The cooperation level of the network playing Snowdrift Game reduces correspondingly, although it is almost invisible. In particular, there exists an optimal intermediate region of network interdependence maximizing the growth rate of the evolution of cooperation on the network playing Prisoner’s Dilemma. Remarkably, players contacting with other network have advantage in the evolution of cooperation than the others on the same network. Introduction. – The problem of cooperation represents a social dilemma characterized by the conflict of interest between group and individuals. The understanding of emergence and maintenance of cooperation among unrelated individuals in real world is still one of the most challenging problems in social and biological systems. Evolutionary game theory provides a uniform mathematical framework to deal with this challenge [1, 2]. The Prisoner’s Dilemma (PD) and Snowdrift Game (SG) are considered as two of the most typical paradigms of investigating cooperation in social dilemmas. They are both twoperson games, where cooperators are prone to exploitation by defectors and the accumulated payoffs of populations are lower than that of pure cooperators [3]. In conventional forms, both PD and SG are played by two individuals deciding simultaneously whether to cooperate or defect. Players both receive R by mutual cooperation, whereas mutual defection results in payoff P for both of them. The highest payoff T is obtained by a player defecting against a cooperator, while the cooperator bearing the cost of S. In the PD, T > R > P > S is achieved. Consequently, it is best for individuals to defect regardless of the co-player’s decision. And, defection is the evolutionarily stable strategy (ESS) in well-mixed infinite populations, even though individuals would be better off if they cooperated [4]. Thus, social dilemma is obvious. While in the SG, payoffs P and S have a reverse order, T > R > S > P , which fundamentally changes the situation and leads to persistence of cooperation. As a result, the replicator dynamics of the SG converges to a mixed stable equilibrium where cooperators and defectors coexist. In this state, the population payoff is smaller than that of full cooperation, hence the SG still represents a social dilemma [5]. Several mechanisms have been proposed in different contexts to elucidate the ubiquitous cooperative behaviors, such as kin selection, direct reciprocity, indirect reciprocity, network reciprocity and group selection [6]. In ad-