## 3 Hierarchical Stackelberg Game Model for Traffic Management  
In Figure 1, vehicles are represented by color-coded blocks at four traffic intersections: North, South, East, and West signals. Traffic flow is organized by designating lanes for left turns (LL), straight movements (SL), and right turns (RL). A red block represents an emergency vehicle positioned in the right lane (RL) at the South signal, indicating that in a real-world scenario, traffic management systems will prioritize its movement to facilitate quick navigation. In this diagram, multiple signals control the flow of vehicles in an urban traffic setting, ensuring that different lanes receive appropriate signals to proceed accordingly. The current model abstracts certain elements of the real-world traffic layout for clarity and computational feasibility but still captures the structural dynamics of signals-controlled urban intersections. This abstraction enables us to focus on assessing the effectiveness of the proposed decision-making mechanism (the hierarchical Stackelberg model) under controlled and interpretable conditions. In addition, the simplified layout retains key features that represent real intersections, such as directional lane and multi-agent interaction. Our main goal at this stage is to validate the fundamental algorithm behavior and assess its efficiency in the process of priority-based scenarios. There are three key components of the Hierarchical Stackelberg Game Model for Traffic Management: the Traffic Management Center (TMC), Emergency Vehicles (EVs), and Regular Vehicles (RVs). A TMC acts as a central authority for regulating and coordinating traffic flow, similar to a traffic control system used by cities. EVs, such as ambulances and fire trucks, are the system's high-priority agents (leaders) that must reach their destinations quickly. Priority is given to these vehicles over regular traffic, which influences optimal routing. RVs, as lower-priority agents (followers), adjust their routes and speeds based on decisions made by emergency vehicles and the TMC, ensuring efficient and coordinated traffic management. With a Stackelberg traffic management system, the Traffic Management Center (TMC) would prioritize certain lanes based on traffic flow. Depending on the signal changes, the vehicles would adjust their actions. Figure 2 depicts a hierarchical Stackelberg game model for traffic management, focusing on the interaction between the Traffic Management Center (TMC), the Emergency Vehicle Agents (EV), and the Regular Vehicle Agents (RV). Two levels of Stackelberg games are represented in the diagram at different levels of the decision-making hierarchy. Similarly, emergency vehicles ensure that regular vehicles can move through traffic efficiently around them. While maintaining a smooth traffic flow, the hierarchical Stackelberg game structure balances the needs of emergency and regular vehicles. By coordinating signal management, such a hierarchical decision-making model can help demonstrate its efficiency. Two levels are involved in the game. At the higher levels, the Traffic Management Center (TMC) acts as the leader, while Emergency Vehicles (EVs) follow. For traffic management modeling, the TMC considers several variables, such as the vehicle's position on the \(x\)- and \(y\)-axes, speed, orientation angle, direction of approach (left, center, right), intended  
Figure 1: Representation of simulated intersection with 3 lanes, where the movements go straight, turn right, and turn left are allowed. Figure 2: Autonomous traffic management system. path (right turn, straight turn, left turn), and intersection branch (North, South, East, West). To minimize EV travel times, the TMC sets dynamic signal timings and route priorities for EVs to ensure they follow the fastest routes. In the lower levels of the game, emergency vehicles (EVs) lead, and regular vehicles (RVs) follow.