Stackelberg leadership model

Authors: Peiying Li (pl634), Xiangyu Zeng (xz853), Wen Su (ws482), Hongyan Ke (hk838), Zhiyu An (za65) (SYSEN 5800/6800 Fall 2024)

Stewards: Nathan Preuss, Wei-Han Chen, Tianqi Xiao, Guoqing Hu

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. 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. His pioneering research emerged during a period when economists were increasingly focused on understanding market dynamics beyond perfect competition and monopoly scenarios. 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.

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 $C_{1},C_{2}$.
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 $Q=q_{1}+q_{2}$.
• Cost functions: $C_{1}(q_{1})$ and $C_{2}(q_{2})$.

Definition

• $q_{1}$: Leader's quantity, chosen to maximize the leader's profit.
• $q_{2}$: Follower's quantity, which depends on the leader’s decision through the reaction function $q_{2}=f(q_{1})$.
• $P$: Market price.

Steps

1. Follower’s Optimization:

The profit of the follower is revenue minus cost. Revenue is the product of price and quantity: $\pi _{2}=P(Q)\cdot q_{2}-C_{2}(q_{2})$, where $\pi _{2}$ denotes the follower’s profit. The best strategy is to find the value of $q_{2}$ that maximizes $\pi _{2}$ given $q_{1}$.

To figure this out, first differentiate $\pi _{2}$ with respect to $q_{2}$ and set it to zero:

${\frac {\partial \pi _{2}}{\partial q_{2}}}={\frac {\partial P(Q)}{\partial q_{2}}}\cdot q_{2}+P(Q)-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}=0$

The value of $q_{2}$ that satisfies the equation is the best strategy. 2. Leader’s Optimization:

2. Leader’s Optimization:

The profit of the leader is given by:

$\pi _{1}=P(q_{1}+f(q_{1}))\cdot q_{1}-C_{1}(q_{1})$

Where $q_{2}(q_{1})$ is the follower’s quantity as a function of the leader’s quantity.

The optimal strategy is to find the value of $q_{1}$ that maximizes $\pi _{1}$, given $q_{2}=f(q_{1})$. To get the optimal solution, differentiate $\pi _{1}$ with respect to $q_{1}$:

${\frac {\partial \pi _{1}}{\partial q_{1}}}={\frac {\partial P(Q)}{\partial q_{2}}}\cdot {\frac {\partial f(q_{1})}{\partial q_{1}}}\cdot q_{1}+{\frac {\partial P(q_{1}+q_{2})}{\partial q_{1}}}\cdot q_{1}+P(q_{1}+f(q_{1}))-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}=0$

Solve $\max _{q_{1}}\pi _{1}$ to find the optimal $q_{1}^{*}$.

3. Compute Follower’s Output:

Plug $q_{1}^{*}$ into $q_{2}=f(q_{1})$ to find the optimal $q_{2}^{*}$.

4. Market Price:

Calculate $p$

5. Output:

• Leader’s output: $q_{1}^{*}$
• Follower’s output: $q_{2}^{*}$
• Market price: $P$

Numerical Examples

In this section, a numerical example is present to illustrate how the Stackelberg leadership model functions. This example demonstrates the strategic decision-making process of two firms in a market where one firm acts as the leader and the other as the follower.

Problem Description

Scenario: Two firms, Firm L (the leader) and Firm F (the follower), produce a homogeneous product and compete in the same market. Firm L moves first by choosing its production quantity $q_{L}$. Firm F observes $q_{L}$ and then decides its own production quantity $q_{F}$. The market price depends on the total quantity supplied by both firms.

Variables and Parameters

Variables:

• $q_{L}$: Quantity produced by the leader (Firm L).
• $q_{F}$: Quantity produced by the follower (Firm F).
• $Q$: Total market quantity, where $Q=q_{L}+q_{F}$.
• $P$: Market price as a function of total quantity $Q$.
• $\pi _{L}$: Profit of the leader.
• $\pi _{F}$: Profit of the follower.

Parameters:

• $a$: Maximum price consumers are willing to pay when the quantity is zero ($a=\$150$).
• $b$: Rate at which the price decreases as quantity increases ($b=\$3$).
• $c_{L}$: Constant marginal cost for the leader ($c_{L}=\$30$ per unit).
• $c_{F}$: Constant marginal cost for the follower ($c_{F}=\$20$ per unit).

Objective Functions and Constraints

Inverse Demand Function: $P(Q)=a-bQ$

Leader’s Objective Function: $\pi _{L}=P(Q)\cdot q_{L}-c_{L}q_{L}$

Follower’s Objective Function: $\pi _{F}=P(Q)\cdot q_{F}-c_{F}q_{F}$

Constraints and Assumptions:

• Both firms aim to maximize their respective profits.
• The follower observes the leader’s output before deciding its own.
• Marginal costs are constant and known to both firms.
• The market is perfectly competitive beyond these two firms.

Step-by-Step Solution

Step 1: Follower’s Optimization

The follower decides $q_{F}$ to maximize $\pi _{F}$ after observing $q_{L}$.

Follower’s Profit Function:

$\pi _{F}=[a-b(q_{L}+q_{F})]q_{F}-c_{F}q_{F}$

Substitute the Known Values:

$\pi _{F}=[150-3(q_{L}+q_{F})]q_{F}-20q_{F}$

Simplify:

$\pi _{F}=130q_{F}-3q_{F}q_{L}-3q_{F}^{2}$

First-Order Condition (FOC):

Differentiate $\pi _{F}$ with respect to $q_{F}$ and set it equal to zero:

${\frac {\partial \pi _{F}}{\partial q_{F}}}=130-3q_{L}-6q_{F}=0$

Solve for $q_{F}$:

$q_{F}=21.67-0.5q_{L}$

This is the Follower’s Reaction Function.

Step 2: Leader’s Optimization

The leader anticipates the follower’s reaction and chooses $q_{L}$ to maximize $\pi _{L}$.

Substitute the Follower’s Reaction Function into the Leader’s Profit Function:

$\pi _{L}=[a-b(q_{L}+q_{F})]q_{L}-c_{L}q_{L}$

Substitute $q_{F}=21.67-0.5q_{L}$:

1. Total Quantity:

$Q=q_{L}+q_{F}=q_{L}+21.67-0.5q_{L}=(1-0.5)q_{L}+21.67=0.5q_{L}+21.67$

2. Market Price:

$P(Q)=150-3Q=150-3(0.5q_{L}+21.67)=150-1.5q_{L}-65=85-1.5q_{L}$

3. Leader’s Profit Function:

$\pi _{L}=(85-1.5q_{L})q_{L}-30q_{L}=85q_{L}-1.5q_{L}^{2}-30q_{L}=55q_{L}-1.5q_{L}^{2}$

First-Order Condition (FOC):

Differentiate $\pi _{L}$ with respect to $q_{L}$ and set it equal to zero:

${\frac {d\pi _{L}}{dq_{L}}}=55-3q_{L}=0$

Solve for $q_{L}^{*}$:

$q_{L}^{*}={\frac {55}{3}}\approx 18.33$

Step 3: Compute Follower’s Optimal Output

Substitute $q_{L}^{*}$ back into the follower’s reaction function:

$q_{F}^{*}=21.67-0.5\times 18.33=21.67-9.17\approx 12.50$

Step 4: Determine Market Price

Calculate total quantity and market price:

1. Total Quantity:

$Q^{*}=q_{L}^{*}+q_{F}^{*}=18.33+12.50=30.83$

2. Market Price:

$P^{*}=150-3\times Q^{*}=150-3\times 30.83=150-92.50\approx 57.50$

Step 5: Calculate Profits

Leader’s Profit:

$\pi _{L}^{*}=P^{*}q_{L}^{*}-c_{L}q_{L}^{*}=57.50\times 18.33-30\times 18.33\approx 504.08$

Follower’s Profit:

$\pi _{F}^{*}=P^{*}q_{F}^{*}-c_{F}q_{F}^{*}=57.50\times 12.50-20\times 12.50\approx 468.75$

Pseudocode Demonstration

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Interpretation

• Leader’s Advantage: By moving first, the leader produces a larger quantity and secures a higher profit than the follower.
• Follower’s Response: The follower adjusts its production based on the leader’s output to maximize its own profit.
• Market Impact: The leader’s strategic decision influences the market price and the total quantity supplied, demonstrating the importance of first-mover advantage in the Stackelberg model.

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.

Case Study

The Stackelberg leadership model has proven to be a versatile tool for addressing hierarchical optimization problems across a wide range of domains, where decision-making processes are structured around a leader-follower dynamic. In the realm of supply chains, Das et al. (2021) explored pricing and ordering strategies within a two-echelon structure under a price discount policy, using the Stackelberg framework to optimize profitability and coordination between manufacturers and retailers. Similarly, Wang et al. (2016) applied the model to product family architecture design, illustrating how leaders' decisions on product design influence supply chain efficiency and profitability. Beyond supply chains, Mandel and García-Alvarado (2019) examined the global implications of tax evasion using the Stackelberg model, uncovering the dynamics of the worldwide network of tax evasion and offering strategies to mitigate its impact on international economies. Expanding into energy systems, Gao et al. (2023) introduced a bi-level multi-leader multi-follower Stackelberg game model for optimizing multi-energy retail packages, effectively balancing energy pricing strategies to enhance both profitability and consumer satisfaction in modern power systems.

In urban IoT networks, Jin et al. (2024) applied the Stackelberg leadership model to optimize pricing strategies for electric vehicle (EV) charging stations. In this context, charging station managers act as leaders, setting service prices to maximize profits, while EV users, as followers, select charging stations based on price, congestion levels, and spatial proximity to minimize their costs. To tackle the intricate interplay between pricing strategies and user decisions, the authors developed the Segmentation-Based Pricing with Iterative Optimization (SPITER) algorithm, incorporating congestion effects and geographic distributions. Real-world evaluations demonstrated that this approach significantly enhances both station profitability and user satisfaction, highlighting the potential of the Stackelberg framework to support urban planners in optimizing resource utilization and improving user experience.

The impact of the Stackelberg leadership model extends further into security domains, as highlighted in Sinha et al.’s paper "Stackelberg Security Games: Looking Beyond a Decade of Success." The Stackelberg Security Games (SSG) framework has been pivotal in optimizing resource allocation across areas such as infrastructure protection, wildlife conservation, cybersecurity, and maritime security. Notable implementations include the ARMOR system at Los Angeles International Airport, which randomized vehicle checkpoint schedules to deter attacks, and the PROTECT system deployed by the U.S. Coast Guard to enhance patrol effectiveness in critical ports. In wildlife conservation, the PAWS system improved ranger patrols to combat poaching in Uganda and Malaysia. The SSG framework’s ability to incorporate adversary behavior and prioritize high-value targets under resource constraints has demonstrated its effectiveness in real-world scenarios. Extensions into emerging areas, such as privacy auditing and e-commerce fraud prevention, further underscore its potential for broader applications. These examples highlight the transformative role of the Stackelberg model in advancing data-driven decision-making and addressing complex, multi-faceted challenges.

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.