A 2×2 game model implemented by co-evolution of both networks and strategies Is established. Several numerical experiments considering various 2×2 game classes, including Prisoner's Dilemma (PD), Chicken, Leader, and Hero, reveal that the proposed co-evolution mechanism can solve dilemmas in the PD game class. The result of solving a dilemma is the development of mutual-cooperation reciprocity (R reciprocity), which arises through the emergence of several cooperative hub agents, which have many links in a heterogeneous and assortative social network. However, the co-evolution mechanism seems counterproductive in case of the Leader and Hero game classes, where alternating reciprocity (ST reciprocity) is more demanding. It is also suggested that the assortative and cluster coefficients of a network affect the emergence of cooperation for R reciprocity.