# Cooperation in Social Dilemmas: A Group Game Model with Double-Layer Networks
## 1 Introduction  
Complex networks can productively depict different systems in society and nature [1], in which the edges of the network represent the relationships between systems. Moreover, interactions are often multidimensional, for example, people are sometimes in multiple social networks [2] and ecosystems interact in various ways at the same time [3]. Therefore, many researchers in the field of complex network research have gradually begun to focus on multi-layer networks instead of single-layer networks. Accordingly, many research results may be applied to everyday life. For instance, the discovery and utilization of multi-layer social networks [4], communication networks [5], and satellite networks [6], etc. To summarize, multi-layer network research is crucial for development in many fields. A good relationship exists between game theory and complex networks, since cooperation is the cornerstone of human development [7, 8]. Accordingly, cooperation among rational individuals can be explored thanks to the discovery of game theory. Meanwhile, complex networks provide a tool for describing game relationships between individuals. Typical game models include the prisoner's dilemma [9], snowdrift game [10], and stag hunting game [11]. The key information of a game includes the participants, strategies, and payoffs. The method of strategy learning directly affects the payoff of the game and the situation of cooperation. Thus, it is very important to choose an appropriate strategy learning method. Based on the above, in recent years, an increasing number of researchers have used a combination of complex networks and game theory to explain the evolution of abundant complex systems, especially in terms of the exploration of the emergence of cooperativebehavior [12]; however, in previous studies, most evolutionary games are based on one-dimensional networks. Recently, research has shifted from a single isolated network to an interdependent double-layer network. Moreover, the previous research with game theory has not considered personal characteristics. In fact, game behaviors will be affected by personal attributes in a practical scenario. In order to solve these problems and attract richer research, this paper proposes a \"public goods game with selection preferences\" (PGG-SP) model that uses a double-layer complex network that is based on reality, in which the double-layer theoretical framework is paired with evolutionary multiplayer game theory. We find that human strategies are composed of long-term selection tendencies and momentary decisions. Therefore, this paper incorporates individual preference attributes into the learning rules for game strategies. The double-layer complex network can usefully illustrate the multiple identities of individuals in the real world. Subsequently, the PGG-SP model can beneficially express the evolution of human strategies in scenarios involving conflicts between collective and personal interests. In short, this model can explain the cooperative evolution problems faced by human beings at multiple levels of social life at the same time in terms of the pertinent influences. In addition, we further explore the effect of networks and game strategies in terms of co-evolution for cooperation. Our results provide a basis for making better human decisions when facing social dilemmas. The main contributions of this article are summarized as follows:  
1. We merge an improved group game with double-layer networks and demonstrate the evolution of cooperation in the double-layer networks based on small-world and scale-free networks. By comparing the evolution behavior of single- and double-layer networks, it is proven that double-layer networks can promote cooperation in multiplayer games. Subsequently, we further discuss the reasons for this phenomenon and the roles of the important parameters in the network. 2.