Dynamics of cooperative evolution with leader-follower hierarchy under information uncertainty in two-layer grid networks

Abstract

Drawing on the idea of leader-follower game theory, this paper investigates the evolution of cooperative behavior in two-layer grid networks. The two-layer network is divided into a leader layer (upper layer) and a follower layer (lower layer), where the snowdrift game (SDG) and the prisoner’s dilemma game (PDG) are conducted, respectively. Based on an updating mechanism of the modified Fermi probability function that incorporates strategy alignment, the leader layer first completes strategy update, and the follower layer then updates its strategy after acquiring information from the upper layer via perception probability. Numerical simulations reveal that the cost-benefit ratio significantly influences the differences in perception strategies between layers. Specifically, at lower ratios, there is a positive correlation between followers’ cooperation and their perception probability, whereas leaders exhibit a negative correlation. The opposite trends are observed at higher ratios. Compared with single-layer networks, the leader-follower structure with two-layer networks significantly enhances the cooperation in the follower layer and also lightly fosters the cooperation in leader layer. Additionally, factors such as inter-layer coupling strength, rewards and punishments for strategy consistency, and selection intensity all promote the emergence of cooperation to varying degrees. This leader-follower hierarchical game offers a new research paradigm for interaction mechanisms in complex social systems.