## 1 Introduction
The proposed hierarchical Stackelberg Fairness Model addresses these limitations directly. It introduces a multi-tier decision-making framework that separates the strategic role of the Traffic Management Center (TMC) from the individual vehicle level. By incorporating fairness constraints measured with Gini coefficients and skewness into the optimization process, the model guarantees fair results. In addition, it includes a dynamic priority processing system that allows emergency vehicles (EVs) to adjust their route in real time while maintaining the regular vehicles (RVs)' balance flow. This structure improves both scalability and adaptability, enabling effective performance at different traffic densities and operational demands. The paper proposes a hierarchical Stackelberg model for traffic management, which extends the traditional framework with multiple decision-making levels. Using this model, the Traffic Management Center (TMC) sets traffic signals and routing priorities as the top-level leader. A first-level follower is an emergency vehicle that receives priority within the network, while a second-level follower is a regular vehicle that adjusts their route according to emergency vehicle decisions and traffic control strategies. As a result of this hierarchical structure, we can control traffic more precisely, particularly in scenarios where emergency vehicles deserve priority while minimizing the delays of regular traffic and ensuring fairness. This is the first study to explore traffic management using a hierarchical Stackelberg approach. In previous studies, the Stackelberg model was used mainly without indicating vehicle types or traffic flows. On the contrary, our hierarchical approach allows more flexibility and efficiency in dealing with complex traffic situations, even in highly congested urban areas where different levels of traffic priority must be balanced. Previous studies have widely adopted [PERSON]'s formulations for traffic management, including the optimization of isolated intersections [2], deep- reinforcement learning for signal control [3], and urban implementation in the real world [4]. Other works include autonomous vehicle control [6] and vehicle racing scenarios [7] in game theoretical frameworks. However, none of these studies implemented a hierarchical Stackelberg architecture that explicitly separated the decision-making layers (such as traffic management centers and individual agents), integrated fairness metrics, and dynamically reconfigured signal priorities in emergency and congestion scenarios. To the best of our knowledge, this is the first study to develop urban traffic control using a Hierarchical Stackelberg game incorporating multi-agent fairness constraints and real-time emergency responses. Based on our simulations, we demonstrate that the Hierarchical Stackelberg model outperforms conventional models in terms of reduced travel times, shorter queues, and improved fairness in traffic management. 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. As for the rest of this paper, it is organized as follows: Section 2 provides a review of related literature on traffic control and game theory approaches. In Section 3, we explain the theoretical foundation for our Hierarchical Stackelberg model. In Section 4, the experimental setup is described, the results are discussed, and the proposed model is compared with existing approaches.