Evolution of cooperation in a mixed cooperative–competitive structured population

Cooperation and competition are two pivotal topics in the literature on the evolutionary dynamics of individual behavior on social networks. This study provides a perspective of joint analysis of social dilemmas within groups and inter-group competition in a mixed cooperative–competitive structured population. Although specific mechanisms for interpreting the emergence and promotion of cooperation have been proposed, including reward, punishment, reputation, and environmental factors, little is known about how inter-group competition affects the cooperation level of groups, especially the intensity and structures of competitive games. Based on multi-games, a mixed cooperative–competitive game model is proposed that individuals within a group engage in a donation game and those from different groups participate in competitive games. The results of numerical simulations suggest that the reward of inter-group competition serves as an external incentive, fostering cooperation within groups. As the intensity of competitive games increases, cooperation within groups is prompted. Additionally, strategic inter-group connections based on degree heterogeneity significantly enhance within-group cooperation, particularly when hubs from one group engage in competitive games with marginal nodes from another group. This research contributes valuable insights into the dynamics of cooperation within groups considering the impact of inter-group competition on within-group cooperation.