Stackelberg Population Dynamics: A Predictive-Sensitivity Approach

Abstract

Hierarchical decision-making processes traditionally modeled as bilevel optimization problems are widespread in modern engineering and social systems. In this work, we deal with a leader with a population of followers in a hierarchical order of play. In general, this problem can be modeled as a leader–follower Stackelberg equilibrium problem using a mathematical program with equilibrium constraints. We propose two interconnected dynamical systems to dynamically solve a bilevel optimization problem between a leader and follower population in a single time scale by a predictive-sensitivity conditioning interconnection. For the leader’s optimization problem, we developed a gradient descent algorithm based on the total derivative, and for the followers’ optimization problem, we used the population dynamics framework to model a population of interacting strategic agents. We extended the concept of the Stackelberg population equilibrium to the differential Stackelberg population equilibrium for population dynamics. Theoretical guarantees for the stability of the proposed Stackelberg population learning dynamics are presented. Finally, a distributed energy resource coordination problem is solved via pricing dynamics based on the proposed approach. Some simulation experiments are presented to illustrate the effectiveness of the framework.

Approximate Sensitivity Conditioning and Singular Perturbation Analysis for Power Converters

“…A prediction-correction scheme for solving convex optimization problems with time varying objective and constraint functions was proposed in [35]. Lastly, in [36], sensitivity conditioning is used to solve a bi-level optimization problem that includes a leader and follower population, obtaining what is known as the Stackelberg equilibrium.…”