The article explores how to solve complex problems involving optimal control and game theory on networks. The researchers developed equations to describe these situations, considering different costs and dynamics in each part of the network. They found unique solutions for these equations, proving their existence and uniqueness under specific conditions. The study also delved into stochastic games on networks, showing how players' behaviors can be modeled and analyzed in these settings. Overall, the research sheds light on how to tackle intricate decision-making processes in networked systems.