Cooperative Intelligent Transport Systems (C-ITS) address contemporary transportation challenges, as Connected Vehicles (CVs) can play a pivotal role in enhancing efficiency and safety. The role of central governing authorities in shaping traffic management policies for CVs influences decision-making processes and system performance. In this work, the role of central governing authorities in the traffic management of a mixed traffic network is examined, integrating System Optimum principles with game theory. More specifically, we introduce and develop a framework that models and analyses the strategic interactions between different stakeholders in a mixed traffic environment, considering central governing authorities with varying levels of control. The results indicate how the various levels of control of a central governing authority may have an impact on the network in terms of traffic measures. Through a strategic games analysis, the trade-offs associated with centralized control mechanisms are demonstrated and recommendations are offered for policymakers and practitioners to optimize traffic management strategies.