  Motivated by the omnipresence of hierarchical structures in many real-world
applications, this study delves into the intricate realm of bi-level games,
with a specific focus on exploring local Stackelberg equilibria as a solution
concept. While existing literature offers various methods tailored to specific
game structures featuring one leader and multiple followers, a comprehensive
framework providing formal convergence guarantees to a local Stackelberg
equilibrium appears to be lacking. Drawing inspiration from sensitivity results
for nonlinear programs and guided by the imperative to maintain scalability and
preserve agent privacy, we propose a decentralized approach based on the
projected gradient descent with the Armijo stepsize rule. The main challenge
here lies in assuring the existence and well-posedness of Jacobians that
describe the leader's decision's influence on the achieved equilibrium of the
followers. By meticulous tracking of the Implicit Function Theorem requirements
at each iteration, we establish formal convergence guarantees to a local
Stackelberg equilibrium for a broad class of bi-level games. Building on our
prior work on quadratic aggregative Stackelberg games, we also introduce a
decentralized warm-start procedure based on the consensus alternating direction
method of multipliers addressing the previously reported initialization issues.
Finally, we provide empirical validation through two case studies in smart
mobility, showcasing the effectiveness of our general method in handling
general convex constraints, and the effectiveness of its extension in tackling
initialization issues.
