Three-Tier Stackelberg Game Model

• Three-tier Stackelberg game model is a hierarchical framework where leaders, intermediates, and followers make sequential decisions to optimize nested objectives.
• It employs nested optimization and mathematical programming with complementarity constraints to solve equilibrium strategies under varying complexity.
• Applications include supply chain management, wireless communications, and data markets, with robust control and incentive design addressing time-inconsistency.

A three-tier Stackelberg game model is a hierarchical, multi-level generalization of the classical leader-follower Stackelberg framework, in which three classes of agents—each occupying a distinct level—interact sequentially by hierarchical decision-making and strategic optimization. The structure may involve explicit leader–follower–follower (or leader–intermediate–follower) roles; extensions may incorporate additional mediators, constraints, or informatics structures such as asymmetric observation. In three-tier Stackelberg games, the analysis centers on determining multi-level equilibrium strategies given the anticipatory reaction functions at each level, integrating a sequence of nested optimization problems, and, in advanced variants, addressing practical, informational, or enforcement complications unique to the triple hierarchy.

1. Fundamental Structure and Mathematical Formulation

A canonical three-tier Stackelberg game comprises three sets of agents, denoted as Level-1 (leader), Level-2 (intermediate leader/follower), and Level-3 (bottom-level follower), with each player indexed by their level. Formally, with decisions $x_1, x_2, x_3$, objectives $f^\ell$, and feasibility sets $C^\ell$, the equilibrium is given via a system of nested optimization problems, typically:

• Level 3 (follower): $$x_3^*(x_1,x_2) \in \arg\min_{x_3} f^3(x_1, x_2, x_3),\ g^3(x_1,x_2,x_3) \ge 0$$
• Level 2 (intermediate): $$x_2^*(x_1) \in \arg\min_{x_2} f^2(x_1, x_2, x_3^*(x_1, x_2)),\ g^2(x_1,x_2,x_3^*(x_1,x_2)) \ge 0$$
• Level 1 (leader): $$x_1^* \in \arg\min_{x_1} f^1(x_1, x_2^*(x_1), x_3^*(x_1, x_2^*(x_1))),\ g^1(\ldots) \ge 0$$

This nested structure is equivalent to a trilevel mathematical programming problem and can be recast using mathematical programs with complementarity constraints (MPCC) for KKT-based necessary conditions (Koirala et al., 2023).

The game may be static (decisions taken once) or dynamic (controls and states evolve over time), deterministic or stochastic, with further complexity introduced by information asymmetry, coupling constraints, or large-population (mean-field) effects.

2. Model Variants and Application Domains

a) Classical Three-Tier Stackelberg Games

In supply chain management, multi-period discrete Stackelberg models with supplier–manufacturer–retailer structures have been developed, where each tier's profit or cost function depends on own investments and state variables, such as accumulated CSR (Khademi et al., 2015, Khademi et al., 2015). Here, Stackelberg equilibria are characterized by temporally nested Hamiltonians and forward-backward co-state recursions.

In wireless communications, Stackelberg games organize network operators (leaders), content providers (intermediate), and subscribers (followers) for resource pricing and allocation (Xiong et al., 2018). The model supports both continuous- and discrete-strategy spaces and exploits backward induction for subgame-perfect equilibrium computation.

b) Stochastic and Mean-Field Extensions

Recent work considers stochastic dynamics and major-minor agent classes: one major leader, $N_1$ minor leaders, and $N_f$ minor followers (Si et al., 2019). State evolution is modeled via coupled SDEs with mean-field coupling, and equilibrium is defined as an $\varepsilon$-Stackelberg–Nash–Cournot equilibrium, with feedback strategies derived from solutions of high-dimensional forward–backward stochastic differential equations (FBSDEs) and associated Riccati equations.

In data markets and fog computing, the three-tier Stackelberg paradigm covers strategic decision-making among buyers (leaders), brokers/intermediaries (intermediate), and sellers/resources owners (followers), often employing mean-field game theory for scalable analysis and control synthesis (Bo et al., 25 Dec 2025, Zhang et al., 2017).

c) Information Asymmetry and Incentive Design

In models with asymmetric information, filtrations and observable sub-$\sigma$-algebras differ across levels, inducing a hierarchy based not only on strategic power but also on available information (Kang et al., 2022). Solutions require stochastic maximum principle, filter-based Riccati equations, and multi-level FBSDE systems.

Robust control constraints (e.g., $H_\infty$) introduce further structure, such that leaders must design incentives (affine mappings or contracts) that guarantee robust team-optimal solutions under worst-case disturbances while sustaining Stackelberg equilibria at all levels (Xiang et al., 2024).

3. Equilibrium Concepts and Solution Methodologies

The three-tier Stackelberg equilibrium generalizes backward-induction reasoning, requiring:

• Solving the bottom-level (Level-3) problem (possibly Nash among many followers; often via best response or mean-field consistency).
• Substituting up to Level-2, anticipating Level-3 reactions (Nash/Stackelberg).
• Solving Level-1 given the double-nested reactions, yielding an overall equilibrium.

For deterministic models, the equilibrium can be encoded through systems of first-order conditions; in stochastic and/or dynamic models, equilibrium strategies are derived via coupled FBSDEs and feedback via Riccati equations (Si et al., 2019, Kang et al., 2022, Xiang et al., 2024).

General-purpose algorithms for this multilevel optimization—where closed-form solutions are infeasible—include Monte Carlo Multilevel Optimization (MCMO), a sampling-based, derivative-free stochastic search that recursively samples and evaluates lower-level reactions for candidate upper-level decisions (Koirala et al., 2023). This approach is compatible with generic constraints and nonconvexities and, under uniqueness assumptions and infinite sampling, enjoys asymptotic convergence to a global equilibrium.

4. Time-Inconsistency and the Role of Third Parties

A consistent challenge in dynamic three-tier Stackelberg games is time-inconsistency—the phenomenon where initial (open-loop) Stackelberg strategies become suboptimal ex post, as leaders may benefit by deviating from announced plans after followers have responded (Jiang et al., 2024).

To enforce credible equilibrium paths, regulatory/third-party mechanisms are introduced:

• The third party collects deposits/fees ex ante.
• It monitors leader actions, instantly applies penalizing "discounts" (multiplicative reductions in future payoffs) upon deviation.
• Discount rules are model-specific: linear in discrete time, exponential or convex in continuous/mean-field settings.
• For large enough penalty intensity (discount factor), no-defection (no rational incentive for deviation by leaders) conditions can be explicitly constructed.

This regulatory contract approach is technically robust—more flexible than full subgame-perfect commitments or delayed punishment by followers, directly manipulating the leader’s effective discount rate and thus restoring sustainability of the Stackelberg solution (Jiang et al., 2024).

5. Computational Methods and Scalability

Solution methods for three-tier Stackelberg games are fundamentally more complex than bilevel cases, due to the proliferation of nested optimization and complementarity constraints, as well as the possible lack of convexity. The key computational approaches are:

Methodology	Main Features	Notable References
Block-structured Riccati/FBSDE	Closed-form, feedback strategies for LQ models with full information	(Kang et al., 2022, Si et al., 2019, Xiang et al., 2024)
Subgradient/backward induction	Iterative update for discrete-strategy or Nash–Stackelberg games	(Xiong et al., 2018, Khademi et al., 2015, Khademi et al., 2015)
Matching/stable marriage	Many-to-many pairing for resource allocation among tiers	(Zhang et al., 2017)
Monte Carlo Multilevel (MCMO)	Gradient-free, sampling algorithm for multilevel nonconvex/nonsmooth games	(Koirala et al., 2023)

The computational cost of naive enumeration grows exponentially in the number of levels. MCMO leverages local search and optimal lower-level routines (using, e.g., IPOPT for the terminal player) to control complexity. Empirical results indicate that, for trilevel cases, high-quality solutions are achieved within practical runtimes, outperforming generic metaheuristics in certain structured applications (Koirala et al., 2023).

6. Theoretical Characterizations and Practical Insights

Three-tier Stackelberg models have generated comprehensive theoretical characterizations for their equilibria:

• Existence and uniqueness often track to classical concavity/convexity assumptions, properties of the Riccati ODEs, and strict diagonal convexity of payoff functions.
• Feedback law characterizations (via Riccati equations) are available for quadratic cases and models with mean-field or multiple minor agents (Si et al., 2019, Kang et al., 2022, Xiang et al., 2024).
• Subgame-perfect or incentive-compatible equilibrium requires mechanism design or third-party intervention to counteract time-inconsistency (Jiang et al., 2024).
• Mean-field and large-scale game models facilitate approximate Nash–Stackelberg equilibria (of order $O(n^{-1})$ to $O(n^{-1/2})$) in large populations (Bo et al., 25 Dec 2025, Si et al., 2019).

Practically, these results enable implementable, scalable coordination mechanisms in engineered systems (IoT/fog platforms, data markets), supply chains, and robust economic design, with regulatory design emerging as a flexible and effective equilibrium enforcement paradigm. Across models, robust third-party contracts are demonstrably maneuverable in practice, requiring only ex ante deposits and immediate, irreversible enforcement on defection (Jiang et al., 2024).

7. Key Open Challenges and Future Directions

Despite substantial progress, the following remain ongoing research directions:

• Handling nonconvex and high-dimensional trilevel Stackelberg games with strong theoretical guarantees for convergence rates.
• Designing incentive-compatible mechanisms in settings with asymmetric information, stochasticity, and strategic uncertainty.
• Extending mean-field/game-theoretic tools to models with inter-level coupling, partial observation, or adaptive population structure.
• Practical design and empirical evaluation of decentralized, distributed solution protocols for trilevel games in large cyber-physical networks.

As the field advances, emphasis is expected to remain on mathematically-sound equilibrium enforcement, scalable algorithmics, and the integration of Stackelberg (hierarchical) game theory with broader concepts from robust control, machine learning, and large-population economics (Jiang et al., 2024, Koirala et al., 2023, Bo et al., 25 Dec 2025, Kang et al., 2022, Xiang et al., 2024, Si et al., 2019, Zhang et al., 2017).