Nested Extremum Seeking Converges to Stackelberg Equilibrium

TL;DR

This paper shows that nested extremum seeking algorithms can be tuned to converge to Stackelberg equilibria instead of Nash equilibria by adjusting their hierarchical scaling parameters.

Contribution

It introduces a method to steer the convergence of nested extremum seeking from Nash to Stackelberg equilibria through parameter scaling, supported by rigorous stability analysis.

Findings

Convergence to Stackelberg equilibrium achieved with modified scaling laws.

Stability proven using Lie-bracket averaging and singular perturbation.

Demonstrated effect with quadratic example and Fish War game.

Abstract

The nested Extremum Seeking (nES) algorithm is a model-free optimization method that has been shown to converge to a neighborhood of a Nash equilibrium. In this work, we demonstrate that the same nES dynamics can instead be made to converge to a neighborhood of a Stackelberg (leader--follower) equilibrium by imposing a different scaling law on the algorithm's design parameters. For the two--level nested case, using Lie--bracket averaging and singular perturbation arguments, we provide a rigorous stability proof showing semi-global practical asymptotic convergence to a Stackelberg equilibrium under appropriate time-scale separation. The results reveal that equilibrium selection, Nash versus Stackelberg, depends not on modifying the closed-loop dynamics, but on the hierarchical scaling of design parameters and the induced time-scale structure. We demonstrate this effect using a simple quadratic example and the canonical Fish War game. The Stackelberg variant of nES provides a model-free framework for hierarchical optimization in multi-time-scale systems, with potential applications in power grids, networked dynamical systems, and tuning of particle accelerators.