Strategic Interactions in Multi-Level Stackelberg Games with Non-Follower Agents and Heterogeneous Leaders

This paper proposes a novel three-level Stackelberg game framework that incorporates heterogeneous leaders and non-follower agents to accurately model congestion-coupled strategic interactions, demonstrating that ignoring non-participating agents leads to distorted equilibrium predictions in systems like EV charging markets.

The Gist

Imagine a busy city where everyone is trying to get somewhere, but the roads are always jammed. Now, imagine trying to figure out the best way to build new gas stations (or in this case, electric vehicle charging stations) in this city.

This paper is about a new way of thinking about that problem. It argues that most current models make a big mistake: they ignore the "background noise" of the city.

Here is the breakdown using simple analogies:

1. The Old Way: The "Two-Player Chess Game"

Traditionally, when experts model these systems, they look at it like a game of chess between two types of players:

• The Leaders (The Shop Owners): These are the companies building the charging stations. They set prices and decide where to build.
• The Followers (The Customers): These are the electric car drivers. They see the prices and the traffic, then choose which station to go to.

The Flaw: In this old model, everyone else on the road—people driving gas cars, delivery trucks, or people just commuting to work who don't use the charging stations—are treated as invisible. The model assumes they are just a static wall of traffic that never changes.

2. The New Insight: The "Dance Floor"

The authors say this is wrong. In reality, the road is more like a crowded dance floor.

• If the Electric Car (EV) drivers crowd into one area to charge, it gets jammed.
• The Gas Car drivers (the Non-Followers) see this jam and say, "Oh, that looks terrible, I'll take a different route."
• But because the Gas Car drivers leave, the traffic pattern changes, which might make the EV drivers' route look better again.

The Key Point: Even though the Gas Car drivers aren't buying electricity or playing the "game" of the charging stations, their decisions to avoid traffic change the outcome for everyone else. They are active participants in the congestion, even if they aren't customers.

3. The New Model: A Three-Layer Cake

To fix this, the authors created a new "Three-Level Stackelberg Game." Think of it like a three-story building where decisions happen from the top down, but the bottom floor pushes back up.

• Level 3 (The Architect - Long Term): A new company wants to enter the market. They are the "Architect." They decide where to build the charging stations. They have to guess: "If I build here, how will the traffic change, and how will the other companies react?"
• Level 2 (The Managers - Short Term): The existing companies (Incumbents) and the new one now compete on price. They set their prices based on where the cars are going. They know that if they are too cheap, too many cars will come, causing a traffic jam that scares everyone away.
• Level 1 (The Crowd - The Reaction): This is where the magic happens.
  • EV Drivers (Followers): They choose a station based on price and traffic.
  • Gas Car Drivers (Non-Followers): They choose a route to avoid traffic.
  • The Interaction: The EV drivers and Gas drivers are mixed together on the road. If the EV drivers clog a street, the Gas drivers leave. If the Gas drivers leave, the street clears up for the EVs. They are constantly reacting to each other.

Why Does This Matter?

If you ignore the Gas Car drivers (the Non-Followers), you might think: "Hey, if I build a charging station here, it will be super profitable!"

But in the real world, because you ignored the Gas drivers, you didn't realize that building there would cause a massive traffic jam. The Gas drivers would flee, but the EV drivers would also get stuck in the gridlock, making the station useless.

The Conclusion:
By including the "background" people (Non-Followers) in the math, the authors show that the best strategy for a company changes completely. You can't just look at your customers; you have to look at the whole ecosystem of traffic.

The Big Picture

This isn't just about electric cars. This logic applies to:

• Cloud Computing: If a new server farm opens, it might slow down the internet for everyone, causing non-paying users to switch providers, which changes the market for the paying users.
• Digital Marketplaces: If a new seller enters a crowded marketplace, they might clog the search results, causing casual browsers to leave, which hurts the established sellers.

In short: You cannot understand a crowded system by only watching the people who are paying you. You have to watch everyone else, too, because their movements change the shape of the room you are standing in.

Technical Summary

Based on the extended abstract provided, here is a detailed technical summary of the paper "Strategic Interactions in Multi-Level Stackelberg Games with Non-Follower Agents and Heterogeneous Leaders."

1. Problem Statement

The paper addresses a critical gap in modeling strategic interactions within congested systems (e.g., transportation, energy, computing). Existing Stackelberg game models typically rely on a restrictive abstraction where:

• Leaders (providers) anticipate the behavior of Followers (users/customers).
• Non-follower agents (background traffic or users not directly participating in the market) are treated as exogenous (fixed background flow) or ignored entirely.

The Core Issue: In congestion-coupled markets, non-follower agents are neither passive nor static. They actively adapt to congestion caused by leaders and followers, and their behavior, in turn, reshapes the congestion patterns faced by strategic agents. Ignoring this bidirectional coupling leads to systematically distorted equilibrium predictions regarding profitability, infrastructure deployment, and competitive intensity.

2. Methodology

The authors propose a novel Three-Level Hierarchical Stackelberg Framework designed to capture the complex interplay between heterogeneous leaders, strategic followers, and non-follower agents.

A. The Three-Level Structure

The framework models sequential decision-making across three distinct levels (illustrated in Figure 1b of the paper):

1. Level 1 (Lowest - Traffic Equilibrium):
   • Agents: Strategic Followers (e.g., EV drivers) and Non-Follower Agents (e.g., non-EV drivers).
   • Mechanism: Both groups select routes and options to minimize individual costs (travel time, delay, price).
   • Coupling: Non-followers adapt to congestion generated by followers (and vice versa). Their routing decisions are endogenous, affecting the travel times and charging choices of the strategic followers.
2. Level 2 (Intermediate - Short-term Competition):
   • Agents: Incumbent Leaders (existing providers).
   • Mechanism: Providers set prices anticipating how drivers (both followers and non-followers) will respond to congestion.
   • Heterogeneity: Incumbents typically have fixed infrastructure and compete only on price.
3. Level 3 (Highest - Long-term Strategy):
   • Agents: The Entrant Leader (new market entrant).
   • Mechanism: The entrant makes long-term infrastructure decisions (e.g., charger placement) while anticipating the downstream pricing responses of incumbents and the resulting equilibrium traffic patterns of all agents.

B. Solution Approach

The model utilizes equilibrium concepts natural to each layer:

• Lower levels are solved using traffic equilibrium concepts (e.g., Wardrop equilibrium) where agents minimize individual costs.
• Higher levels solve optimization problems that explicitly anticipate the induced responses of lower levels.
• The approach avoids restrictive assumptions like agent homogeneity or exogenous traffic, treating congestion as an endogenous outcome shaped by strategic decisions and adaptive behavior.

3. Key Contributions

• Introduction of Non-Follower Agents: The paper formally integrates non-follower agents into Stackelberg games, demonstrating that their adaptive behavior fundamentally alters equilibrium structures even without direct market participation.
• Heterogeneous Leaders: The framework distinguishes between leaders with different decision horizons and action sets (e.g., incumbents with fixed infrastructure vs. entrants making placement decisions).
• Bidirectional Coupling Modeling: It captures the feedback loop where infrastructure decisions influence congestion, which influences competition, which in turn influences traffic distribution, including non-participating agents.
• Generalizability: While instantiated in the EV charging domain, the framework is applicable to a broad class of congestion-coupled multi-agent systems in mobility, energy, and computing.

4. Results and Application (EV Charging Context)

The framework is instantiated in the context of Electric Vehicle (EV) Charging Infrastructure Planning:

• Scenario: A new entrant selects charger locations (Level 3) while incumbents compete on price (Level 2). EV drivers (Followers) and non-EV drivers (Non-Followers) share the road network.
• Findings:
  • Explicitly modeling non-EV traffic (non-followers) qualitatively changes strategic incentives.
  • Ignoring non-followers leads to incorrect predictions about optimal infrastructure placement and pricing strategies.
  • The model reveals that the entrant must anticipate not just the price competition, but the redistribution of both EV and non-EV traffic caused by congestion interactions.
• Comparison: Unlike existing models that examine provider competition OR charging-traffic interaction in isolation (often assuming fixed locations or exogenous traffic), this model unifies them, showing that the interaction between heterogeneous competitors and background traffic is critical for accurate equilibrium analysis.

5. Significance

• Theoretical Advancement: The paper challenges the standard "benign simplification" of ignoring non-participating agents in game theory. It proves that in congestion-coupled systems, these agents are active participants in the equilibrium dynamics.
• Practical Impact: For policymakers and infrastructure planners (specifically in EV charging), the model suggests that ignoring background traffic (non-EVs) can lead to suboptimal or even detrimental infrastructure investments.
• Broader Applicability: The framework provides a robust tool for analyzing any market where resource value is degraded by congestion and where multiple agent types (strategic vs. non-strategic) interact, extending beyond EVs to cloud computing, energy grids, and digital marketplaces.

In summary, this work argues that accurate modeling of strategic interactions in congested environments requires a three-level Stackelberg approach that treats congestion as an endogenous variable shaped by the adaptive behaviors of both market participants and non-participants.