A Regularized Nikaido-Isoda Function Approach to Multi-Leader-Follower Games

Abstract: A multi-leader--follower game (MLFG) is a hierarchical noncooperative game in which leaders compete at the upper level while taking into account the followers' best responses at the lower level. A typical approach to solving the MLFG reformulates it as an equilibrium problem with equilibrium constraints (EPECs) by replacing the lower-level game with its KKT conditions. Another approach, when each follower's response is unique, is to reformulate the MLFG as a Nash equilibrium problem by substituting these response functions into each leader's problem. However, both reformulations may lack scalability since higher-order derivatives may be required when solving the resulting problems. In this paper, we propose a new reformulation of the MLFG by exploiting a regularized Nikaido--Isoda function and approximating the MLFG by a single-level differentiable Nash equilibrium problem with a penalty parameter. The proposed reformulation neither requires derivative information on the followers' game nor assumes convexity of each follower's problem; hence, it can handle a broader class of MLFGs. Under global subanalyticity, we analyze the mathematical relationship between equilibria of the original MLFG and the proposed reformulation.

• The paper introduces a regularized NI function that converts multi-leader–follower games into a single-level Nash equilibrium without requiring derivative or convexity assumptions.
• It establishes equivalence of equilibria in convex settings and provides Hӧlderian error bounds for nonconvex cases, enhancing solution accuracy.
• The approach offers a derivative-free framework for follower problems, broadening the applicability of MLFGs in networked systems and market designs.

A Regularized Nikaido–Isoda Function Approach to Multi-Leader–Follower Games

Introduction and Context

Multi-leader–follower games (MLFGs) generalize classic Stackelberg frameworks to domains with several interacting leaders and followers, requiring sophisticated treatments for hierarchical noncooperative decision-making. Traditionally, analytical and computational strategies for MLFGs rely on bilevel optimization reformulations, most notably through equilibrium problems with equilibrium constraints (EPECs) or via best-response mappings that assume uniqueness of the follower equilibrium. These formulations tend to become computationally expensive and structurally restrictive, particularly due to their reliance on derivative information, assumptions of convexity, and higher-order smoothness.

This paper introduces a robust alternative: a reformulation of MLFGs grounded in regularized Nikaido–Isoda (NI) functions. This approach enables conversion of the hierarchical game into a single-level differentiable Nash Equilibrium Problem (NEP) with penalty terms, notably without the requirement for derivative or convexity assumptions regarding the follower-level problems. The method exploits regularization and penalty terms to maintain tractability and solution accuracy, thereby broadening the class of MLFGs that can be effectively analyzed and solved.

Technical Contributions

Reformulation Using Regularized Nikaido–Isoda Functions

The core methodological innovation involves the deployment of a regularized NI function to encapsulate the equilibrium constraints of the follower tier. The NI function for the followers' game is defined as

$$\Psi_\mu(x, y, z) = \sum_{\omega=1}^M [\gamma^\omega(x, y^\omega, y^{-\omega}) - \gamma^\omega(x, z^\omega, y^{-\omega})] - \frac{1}{2\mu}\|y - z\|^2\,,$$

where $\mu > 0$ is a regularization parameter. The corresponding value function $h_\mu(x, y)$ acts as a penalized equilibrium gap.

This construction allows:

• Elimination of explicit equilibrium constraints by embedding them into smooth penalty terms.
• Conversion of the hierarchical system into a single-level Nash game, where each leader optimizes an augmented objective:

$$\Theta^\nu_\mu(w^\nu, x^{-\nu}; \rho) = \theta^\nu(w^\nu, x^{-\nu}) + \rho h_\mu(w^\nu, x^{-\nu}),$$

subject to $w^\nu \in X^\nu \times Y$.

Theoretical Analysis and Equilibrium Relationships

The paper rigorously analyzes the relationship between different equilibrium notions in the original MLFG and solutions to the regularized, penalized single-level NEP:

1. Equivalence under Convexity: When the follower NEP is convex, the solution to the regularized NEP coincides with the original MLFG equilibrium set. In nonconvex regimes, the relaxation is shown to provide a necessary condition for MLFG equilibria.
2. Existence and Approximation Results: The use of regularized NI functions under global subanalyticity assumptions (on the objective and constraint functions) provides guarantees for the existence of solutions and establishes H\"olderian error bounds. These bounds are critical for quantifying the approximation quality of penalized solutions relative to the original MLFG solution set.
3. Variational Equilibria and Stationarity: The connection between stationarity in the regularized single-level NEP and the variational equilibria of the MLFG is formalized. The analysis leverages first-order conditions via generalized equations, showing that under suitable partial calmness (similar in spirit to constraint qualification) the KKT points of the penalized NEP correspond to stationary solutions of the original MLFG.

Computational and Practical Considerations

Key aspects emphasized include:

• Derivative-Free Handling of Followers: The approach operates without higher-order derivative information about the follower-level problems, making it well-suited to MLFGs with complex, nonsmooth, or black-box followers.
• Compactness and Subanalyticity: Mild structural requirements (compactness, subanalyticity) suffice for the analysis, greatly expanding potential applications.
• Penalty Parameter Tuning: Convergence to true equilibria is controlled via the penalty parameter $\rho$. As $\rho \to \infty$, the solution to the penalized NEP approaches feasibility with respect to the original equilibrium constraint.

Numerical and Strong Claims

• No Derivative or Convexity Requirement: The method is explicitly valid for MLFGs in which the followers' problems are nonconvex and even nonsmooth, broadening the scope beyond prior EPEC and response-based approaches.
• Feasibility and Stationarity Bounds: Under explicit H\"olderian error bounds and Kurdyka–{\L}ojasiewicz-like conditions, the approximation error and feasibility violation can be quantitatively controlled as a function of penalty parameters and problem structure.
• Equilibrium Set Inclusion and Tightness: In convex regimes, equivalence of solution sets is asserted, while in nonconvex cases, the inclusion of the regularized solution set within the original is proved.

Theoretical and Practical Implications

The regularized NI approach provides an extensible and computationally practical pathway for analyzing and solving complex, large-scale MLFGs arising in networked systems, market design, and strategic resource management. The framework is particularly impactful in settings where derivative information is unavailable, or the follower NEP is ill-behaved (e.g., nonconvex payoff landscapes).

On the theoretical front, the results establish firm connections among various equilibrium concepts (Nash, generalized Nash, variational equilibria), unify them under a single-level reformulation framework, and supply the mathematical underpinnings (through subanalyticity and regularization theory) required for robust optimization.

Potential Future Directions

Several avenues emerge:

• Extension to State-Dependent Follower Sets: The current approach assumes fixed feasible sets $Y^\omega$ for followers. Allowing $Y^\omega(x)$ would require intricate set-valued analysis but would accommodate many practically relevant hierarchical problems.
• Algorithmic Enhancements: The differentiability and smoothness properties invite the application of first-order and second-order optimization methods for large-scale MLFGs, warranting further study on practical convergence rates and scalable solvers.
• Relaxing Subanalyticity: Investigating the necessity of subanalyticity and exploring whether similar error bounds can be achieved under weaker or alternative regularity conditions.

Conclusion

By exploiting regularized Nikaido–Isoda functions, this work delivers a flexible, smooth, and computationally tractable reformulation for multi-leader–follower games that avoids restrictive assumptions prevalent in previous formulations. The approach systematically clarifies the relationship between various equilibrium concepts and provides explicit error and feasibility guarantees under mild assumptions. This framework lays the groundwork for further algorithmic development and theoretical exploration in hierarchical games, particularly in nonconvex and derivative-limited environments.