Equilibrium analysis of networked game under best-response dynamics

Both coordination, a behavioral pattern where individuals adopt the same behavior as the majority, and anticoordination, a behavioral pattern where individuals choose a behavior different from the majority, are prevalent in social and engineering scenarios. Examples include the spread of social norms or innovations for coordination and the division of labor and traffic congestion for anti-coordination. Exploring the large-scale interacting systems driven by coordination and anticoordination is of great significance in understanding, predicting, and optimizing the evolutionary outcomes. While existing studies have primarily focused on analyzing the convergence of networked systems and evaluating the time to reach equilibrium, our investigation delves into all possible states of a networked system and scrutinizes all equilibrium states. Notably, we observe that the number of equilibrium states increases exponentially with the network size. Accounting for network structure properties, homogeneous networks exhibit more equilibrium states than heterogeneous networks. Furthermore, for specific behavioral switching thresholds, the number of equilibrium states is surprisingly higher than at neighboring switching thresholds. Our work reveals the intricate interplay between evolutionary dynamics, system structure, and behavioral switching thresholds.