Stackelberg leadership model

Authors: Peiying Li, Xiangyu Zeng, Wen Su, Hongyan Ke, Zhiyu An (SYSEN 5800/6800 Fall 2024)

Introduction

The Stackelberg leadership model, also known as the Stackelberg game or Stackelberg competition, is a strategic game in economics and game theory where one firm (the leader) makes its decisions before other firms (the followers) in an imperfectly competitive market[1]. This sequential decision-making structure fundamentally differs from simultaneous-move games like the Cournot model, as it introduces the element of strategic advantage through first-mover position.

Heinrich von Stackelberg first introduced this concept in 1934 through his work "Market Structure and Equilibrium" (Marktform und Gleichgewicht). His pioneering research emerged during a period when economists were increasingly focused on understanding market dynamics beyond perfect competition and monopoly scenarios[2]. The model addressed a critical gap in economic theory by examining how firms might behave when they make decisions in a sequential order rather than simultaneously[3].

The motivation for studying the Stackelberg model stems from its widespread applicability in real-world markets. Many industries exhibit leader-follower dynamics, where established firms act as market leaders while others respond to their decisions. For instance, in the commercial aircraft manufacturing industry, Boeing and Airbus often demonstrate Stackelberg-like behavior in their capacity decisions and product launches. Similarly, in retail markets, large chains frequently make pricing and inventory decisions that smaller retailers must then react to.

The model serves several key purposes in modern economic analysis:

1. Understanding how sequential decision-making affects market outcomes
2. Analyzing the strategic advantages of being a first mover in a market
3. Predicting price levels and market quantities in leader-follower scenarios
4. Providing insights for regulatory policy in markets with dominant firms

Algorithm Discussion

Condition and Assumption

The Stackelberg Leadership Model consists of sequential decisions by a leader and a follower, optimizing their strategies under specific assumptions.

1. Cost function: Both leader and follower incur constant marginal costs $(C1, C2)$.
2. Rationality: Both leader and follower are profit-maximizing agents.
3. Information transparency: The follower has full knowledge of the leader’s decision.

Algorithm Description

Input:

• Inverse demand function: P(Q), where Q=q1+q2
• Cost functions: C1(q1)and C2(q2).

Definition:

• q1: leader’s quantity which maximizes the leader's profit.
• q2: follower's quantity which depends on the leader’s decision through a reaction function q2 = f(q1).
• P: market price

Step:

1. Follower’s Optimization

Application - Overview

The Stackelberg Leadership Model has been widely applied across various disciplines, including economics, operations research, engineering, and data science. It is particularly useful in situations involving hierarchical decision-making and strategic interactions between leaders and followers.

The model is used in game-theoretic approaches to cybersecurity and military defense strategies. Leaders may set optimal security measures, and followers (e.g., attackers) choose their best response strategies. One example is optimizing resource allocation for defending critical infrastructures against cyber threats.

The model is also extensively used to study market competition scenarios, particularly in oligopolies where a dominant firm (leader) sets prices or quantities that followers (competitors) respond to. Examples include pricing strategies in retail or telecommunications markets; analyzing the behavior of dominant firms like Amazon or Walmart versus smaller competitors.

The Stackelberg model helps optimize decisions in multi-echelon supply chains. Leaders (e.g., manufacturers) decide production quantities, while followers (e.g., retailers) decide pricing or ordering policies. Examples include coordinating inventory and pricing strategies between a supplier and multiple retailers to maximize overall profit.

Governments and energy producers use the Stackelberg model to balance renewable energy investments and pricing policies. Designing subsidy schemes where governments (leaders) incentivize energy producers (followers) to adopt cleaner technologies.

Application - Case Study

The Stackelberg leadership model has been widely applied to hierarchical optimization problems across diverse domains, where decision-making processes are structured in a leader-follower dynamic. Fawer and Von Stengel (1997) employed the model to optimize pricing strategies in supply chains, enabling manufacturers to set prices that maximize their profits while accounting for the reactions of downstream retailers, such as order quantities and inventory levels. Simchi-Levi et al. (2004) extended the model’s application to logistics and transportation problems, particularly in designing cost-effective distribution networks where central planners act as leaders, influencing the operational decisions of local distribution hubs. García-Alvarado et al. (2015) demonstrated its effectiveness in energy markets, where power generation companies optimized electricity prices and output levels, anticipating the responses of consumers and regulatory bodies to ensure market efficiency and sustainability. Wang and Du (2020) integrated the Stackelberg framework into urban traffic management, formulating optimal toll pricing and road usage strategies to reduce congestion while considering the behavioral adaptations of drivers. Despite its success, computational challenges have been noted, particularly in complex or nonconvex formulations. Li and Ng (2018) suggested leveraging advanced decomposition techniques, such as bi-level optimization algorithms, to address these issues and enhance the model’s scalability and efficiency in large-scale applications. These studies highlight the versatility and potential of the Stackelberg leadership model in solving real-world hierarchical decision problems.

In their 2024 study, Jin et al. applied the Stackelberg leadership model to optimize pricing strategies for electric vehicle (EV) charging stations within urban Internet-of-Things (IoT) networks. In this framework, the charging station manager acts as the leader, setting service prices to maximize profits, while EV users, as followers, select charging stations based on factors such as price, congestion levels, and spatial proximity to minimize their costs. To handle the intricate interactions between pricing strategies and user decisions, the authors developed the Segmentation-Based Pricing with Iterative Optimization (SPITER) algorithm, incorporating congestion effects and geographic distributions. Using real-world urban datasets, they demonstrated that this model not only enhances station profitability but also improves user satisfaction. The study underscores the effectiveness of the Stackelberg framework in managing hierarchical decision-making in IoT systems, offering valuable insights for urban planners to balance resource utilization with user experience.

Application - Software

The Stackelberg Leadership Model is supported by a range of software tools and platforms that cater to optimization and game theory problems, making it accessible for researchers and practitioners across various domains. These tools often integrate mathematical programming and simulation capabilities, enabling the modeling of hierarchical decision-making processes and the computation of equilibrium solutions.

MATLAB’s Optimization Toolbox provides robust support for Stackelberg game modeling. It allows users to implement custom algorithms for leader-follower problems, leveraging its built-in solvers for linear, nonlinear, and mixed-integer programming. MATLAB is particularly popular in academic and engineering settings due to its ease of use and extensive documentation.

GAMS is a high-level modeling system specifically designed for mathematical optimization. It supports Stackelberg game formulations by enabling users to model multi-level decision problems. Its compatibility with solvers like CPLEX and Gurobi makes it a preferred choice for complex supply chain and economic models requiring hierarchical optimization.

Python libraries such as Pyomo provide an open-source platform for defining and solving Stackelberg games. Pyomo’s flexibility allows users to model leader-follower interactions with constraints and objectives at multiple levels. When combined with solvers like Gurobi or GLPK, it becomes a powerful tool for handling real-world hierarchical problems. Other Python libraries, like Scipy and nlopt, can also be used for customized implementations, making Python a versatile option for both research and practical applications.

R offers tools like nloptr for nonlinear optimization, which can be adapted to solve Stackelberg models. Its statistical and visualization capabilities make it a compelling choice for analyzing game outcomes and presenting results in data-intensive projects.

Conclusion

The Stackelberg Leadership Model is an important framework for analyzing decision-making in hierarchical systems, offering valuable insights into how leaders and followers interact in competitive environments. By structuring decisions sequentially, the algorithm shows the advantages of moving first and the strategic responses that follow. The model's assumptions such as rationality, transparency, and constant marginal costs create a solid foundation for understanding leader-follower dynamics and their impact on market outcomes.

One of the key takeaways is the strategic advantage held by the leader who influences the follower’s actions the market equilibrium. The numerical example that we showed in this paper demonstrated how the model can predict optimal strategies for firms, including production levels, pricing, and profits. Beyond economics, the model’s applications extend to diverse fields such as supply chain, cybersecurity, and IoT networks.

The model could be extended to address scenarios with more complexity, such as multiple leaders, dynamic decision-making processes, or incomplete information. By combining real-world factors such as uncertainty or behavioral nuances would make the model even more applicable to modern challenges. Leveraging advanced computational tools, such as machine learning and optimization algorithms, could also enhance its accuracy and usability in tackling complex systems.

Therefore, the Stackelberg Leadership Model is a powerful tool for understanding strategic interactions and optimization decisions in hierarchical settings. While the model has proven to be highly effective, there are still potential improvements for expanding its scope and applications, making it an valuable research area for modern economic and strategic analysis.