Entropy-Regularized RL for Stackelberg Games in Dynamic Models

Congde Hu, Danping Li, Lin Xu, Wenying Xu· June 30, 2026 View original

Summary

This paper proposes an entropy-regularized reinforcement learning (ERRL) approach for linear-quadratic Stackelberg differential games (LQ-SDGs) in regime-switching diffusion models. It uses neural networks to solve high-dimensional PDEs and promotes exploratory policies to avoid suboptimal equilibria.

Solving Stackelberg differential games (SDGs), which model hierarchical decision-making in continuous-time stochastic environments, is computationally intensive, especially in high-dimensional systems. Traditional methods like dynamic programming and Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations face significant challenges. This research introduces an entropy-regularized reinforcement learning (ERRL) approach specifically for linear-quadratic SDGs (LQ-SDGs) within a continuous-time diffusion framework that incorporates Markovian regime switching. A key innovation is the derivation of exploratory, weakly-coupled HJBI equations, which, through entropy regularization, encourage stochastic policies that actively avoid suboptimal outcomes. Neural networks are integrated to approximate regime-dependent value functions and efficiently solve the high-dimensional partial differential equations. A novel sampling technique further enhances computational tractability. Numerical results confirm the framework's effectiveness in escaping suboptimal traps via exploratory policies, highlighting the crucial role of entropy regularization and neural network approximations for robust hierarchical decision-making under abrupt environmental shifts.

Why it matters

This framework offers a powerful tool for optimizing hierarchical decision-making in complex, dynamic systems, which is vital for strategic planning, competitive analysis, and resource allocation in industries facing rapid environmental changes. It enables more robust and adaptive strategies.

How to implement this in your domain

  1. 1Apply this ERRL framework to model and optimize hierarchical decision-making in your organization, such as leader-follower dynamics in supply chains or competitive markets.
  2. 2Utilize neural network approximations to solve complex control problems that were previously intractable due to high dimensionality.
  3. 3Explore the benefits of entropy regularization to encourage more robust and exploratory policies in existing reinforcement learning applications.
  4. 4Develop simulation tools that incorporate regime-switching diffusion models to test strategic responses to sudden environmental shifts.

Who benefits

BFSIManufacturingLogisticsEnergy

Key takeaways

  • ERRL provides a robust solution for hierarchical decision-making in dynamic environments.
  • Entropy regularization promotes exploratory policies, helping to avoid suboptimal equilibria.
  • Neural networks efficiently approximate solutions for high-dimensional problems.
  • The framework is effective in regime-switching diffusion models, relevant for abrupt environmental shifts.

Original post by Congde Hu, Danping Li, Lin Xu, Wenying Xu

"arXiv:2606.28671v1 Announce Type: new Abstract: Stackelberg differential games (SDGs) provide a powerful framework for hierarchical decision-making in stochastic and continuous-time environments, yet their solution remains computationally challenging due to the complexity of trad…"

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Originally posted by Congde Hu, Danping Li, Lin Xu, Wenying Xu on X · view source

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