## 1 Introduction  
Traffic management is a challenge for modern cities due to urbanization, which has increased vehicle density and resulted in frequent traffic jams, which negatively affect travel time and transportation efficiency. Traffic systems are complex, especially in urban areas with high traffic volumes, and require sophisticated control mechanisms that adapt dynamically to real-time conditions. With the aim of reducing delays, reducing congestion, and improving traffic flow through intersections, traffic signal optimization and vehicle traffic routing have become major research areas [1]. Stackelberg games, which treat traffic control as a problem of leadership and followers, have been shown to be effective in solving some of these challenges. This model equates central traffic management with a leader,while vehicles act as followers, adjusting routes and behaviors based on traffic signals and conditions set by the leader. Over the past decade, game theory models--especially [PERSON]'s formulations--have gained interest in traffic control due to their ability to model competitive and hierarchical decision-making. [PERSON]. [2] investigated the [PERSON]/[PERSON] Q-learning model for traffic at a single intersection. Although the framework of reinforcement learning has been adapted to dynamic traffic conditions, the model remains limited to isolated small-scale environments and lacks scaling for multi-agent traffic control across cities. Furthermore, the lack of fairness guarantees led to a biased signal allocation, which adversely affected non-prioritized agents. [PERSON]. [3] proposed a deep Q-network (DQN) method for controlling large-scale traffic signaling based on Nash equilibrium. The model addressed the computational flexibility of large networks but neglected priority differentiations such as emergency vehicles (EVs) and did not consider the level of equitable collaboration between agents or across intersections. [PERSON]. [4] used the [PERSON] and [PERSON] formulations on the real road network in Warsaw to verify them based on empirical data. However, the lack of static modeling of driver behavior and hierarchical priority for important agents, such as EVs, has revealed deficiencies in dynamic traffic allocation and fairness maintenance. Ref. [5] introduced the Nash-Stackelberg-Nash (NSN) game to integrate traffic and energy networks. Their model dealt well with system-level coordination but gave priority to economic efficiency over real-time response, making it inappropriate for traffic situations that require rapid reconfiguration, such as emergency situations or peak traffic congestion. Ref. [6] explored various game structures, including Nash, Stackelberg, and hybrid control. Although theoretically robust, this work lacked a comprehensive vehicle priority management strategy and fairness assurance strategy, making it not practical for heterogeneous urban traffic. Ref. [7] developed a Nash-Stackelberg planner that was adapted to multi-vehicle racing. The model has successfully handled aggressive maneuvers and priority-oriented tactics, but it is highly domain-specific and cannot be transferred to urban traffic management, where shared resources and fairness dominate. Ref. [8] presented a strategy-focused interaction algorithm with real traffic data, incorporating Stackelberg logic. Although their approach captures real-world dynamics, the model lacks system-level fairness restrictions and shows unstable behavior under unbalanced demand conditions. Despite significant advances, existing models often fail in several critical areas. Many approaches give priority to efficiency, such as reducing travel time, but do not include fair considerations, leading to unfair access and allocation of resources. Others lack hierarchical differentiation, treat all vehicles the same way, and therefore neglect the priority of emergency routing. In addition, many models show limited scalability and struggle with coordination when facing dynamic, high-load, or emergency conditions.