## 2 Related Work  
In several studies, biologically inspired algorithms have been used to optimize traffic. According to [9], dynamic vehicular traffic control systems can improve traffic management by reducing congestion through ant colony optimization (ACO) and traffic light optimization. Machine learning and graph-based neural networks have been used to improve traffic-flow forecasting. Ref. [10] researched traffic-flow distribution forecasting in transport networks, presenting statistical methods for predicting vehicular distribution. To prevent traffic congestion zones, ref. [11] used a resilience-based approach, integrating vulnerability analysis of transportation systems. However, ref. [12] forecast traffic flow using graph convolutional neural networks (GCNs ) and branch-and-bound optimization, achieving high predictive accuracy, while [13] used spatial-temporal GCNs for real-time traffic forecasting, applying long-term short-term fusion. Through the application of deep learning and graph-based networks, these methods demonstrate the potential of traffic-flow forecasting to improve urban mobility systems through accurate forecasting [14]. Despite the value of accurate forecasting, we make our contribution by adapting traffic signals and routes in real time rather than relying only on predictive models. As a result of incorporating Stackelberg games into traffic control, we ensure a more dynamic and responsive system that can adjust in real time rather than just forecasting. For adaptive traffic signal control, reinforcement learning has emerged as a key technique. A study [15] pioneered the use of deep reinforcement learning to optimize traffic signal timing, minimizing delays and improving the overall flow of traffic. Ref. [16] applied deep reinforcement learning to vehicular networks for traffic light control, further improving efficiency by learning dynamic traffic patterns. In their study of deep learning and discrete reinforcement learning, [17] emphasized the benefits of DRL for determining traffic signals. The study [18] examined reward definitions for reinforcement learning, which are crucial to adaptive signal control. As a result of these contributions, reinforcement learning can be used to create efficient and responsive traffic systems. Although reinforcement learning (RL) has proven effective for adaptive signal control, our model relies on game theory (Stackelberg games). We incorporate hierarchical prioritization into traffic signal control, making the system responsive to emergency and regular vehicles, whereas traditional RL methods only optimize signal timings based on traffic flow. Stackelberg and multi-agent games have been important in the development of traffic management strategies. A study [19] investigated socially aware multi-agent learning, which takes into account vehicle interactions and optimizes traffic outcomes. In [20], graph attention neural networks were applied to multi-agent game abstraction, which made it possible for agents to make more effective decisions together. Ref. [21] also studied consistent neighborhood cognition through cooperative learning in multi-agent reinforcement learning (MARL). As opposed to multi-agent reinforcement learning (MARL), which emphasizes decentralized learning and agent cooperation, our model applies a Hierarchical Stackelberg game structure that explicitly distinguishes between leaders (Traffic Management Centers and Emergency Vehicles) and followers (Regular Vehicles). As opposed to MARL models, which generally lack explicit prioritization mechanisms, this hierarchical structure is better suited for transportation systems where prioritization is crucial, such as emergency vehicle routing. Traffic congestion has also been modeled and resolved using game theory. According to [22], game-theoretic models have been shown to reduce traffic congestion by applying Stackelberg routing to parallel networks. According to [23], noncooperative transportation can be solved by linear generalized Nash games.