Finding a Multiple Follower Stackelberg Equilibrium: A Fully First-Order Method

2025-09-09

AI Summary

This paper addresses the efficient computation of ε-stationary equilibria in multi-follower Stackelberg games, proposing the first fully first-order algorithm with theoretical convergence guarantees. Methodologically, it reformulates the bilevel game as a single-level constrained optimization problem, approximating the leader’s gradient via the Lagrangian function and relying solely on first-order derivative information; follower best responses are approximated via gradient descent, while leader and follower strategies are updated jointly to handle bilevel coupling. Key contributions include: (1) the first convergence guarantee for a fully first-order method in this setting; (2) elimination of second-order derivatives or implicit differentiation, drastically reducing computational complexity; and (3) strong scalability and numerical stability in multi-follower scenarios. Experiments demonstrate superior performance in accuracy, efficiency, and robustness compared to existing approaches.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

In this work, we propose the first fully first-order method to compute an epsilon stationary Stackelberg equilibrium with convergence guarantees. To achieve this, we first reframe the leader follower interaction as single level constrained optimization. Second, we define the Lagrangian and show that it can approximate the leaders gradient in response to the equilibrium reached by followers with only first-order gradient evaluations. These findings suggest a fully first order algorithm that alternates between (i) approximating followers best responses through gradient descent and (ii) updating the leaders strategy via approximating the gradient using Lagrangian.

Problem

• Finding a multiple follower Stackelberg equilibrium solution
• Developing a fully first-order method with convergence guarantees
• Approximating leader's gradient using Lagrangian without second-order information

Innovation

• Reframe leader-follower as constrained optimization
• Define Lagrangian to approximate leader's gradient
• Alternate follower response and leader update