•We study the interactive heterogeneity of edge-strategies in multi-game.•When multiple heterogeneities are combined, the level of cooperation may be reduced.•Players are allowed to adopt different strategies for different neighbors. In real-world social networks, the diversity of social members usually affects the decision-making behaviors in daily human life. Here, to better capture the interaction characteristics of realistic individuals, we explore the evolution of cooperative behaviour in multi-game under interactive heterogeneity based on edge dynamics. Simulation results show that interactive heterogeneity can lead to the formation of larger clusters of cooperators in the network and significantly improve the cooperative behaviour of the group. However, when interactive heterogeneity and network heterogeneity coexist, worse results are obtained.