Computing Algorithm for an Equilibrium of the Generalized Stackelberg Game

TL;DR

This paper introduces a novel methodology for efficiently computing a generalized Stackelberg equilibrium in hierarchical games, transforming complex multi-follower interactions into a solvable single-leader problem with practical applications in EV charging strategies.

Contribution

It provides a general framework for existence and polynomial-time computation of equilibria in generalized Stackelberg games, including a transformation technique and an implicit gradient descent algorithm.

Findings

Equilibrium existence conditions established using variational equilibrium.

Transformation reduces the problem to a simpler 1-1 Stackelberg game.

Effective gradient-based algorithm successfully computes equilibria.

Abstract

The $1-N$ generalized Stackelberg game (single-leader multi-follower game) is intricately intertwined with the interaction between a leader and followers (hierarchical interaction) and the interaction among followers (simultaneous interaction). However, obtaining the optimal strategy of the leader is generally challenging due to the complex interactions among the leader and followers. Here, we propose a general methodology to find a generalized Stackelberg equilibrium of a $1-N$ generalized Stackelberg game. Specifically, we first provide the conditions where a generalized Stackelberg equilibrium always exists using the variational equilibrium concept. Next, to find an equilibrium in polynomial time, we transformed the $1-N$ generalized Stackelberg game into a $1-1$ Stackelberg game whose Stackelberg equilibrium is identical to that of the original. Finally, we propose an effective computation procedure based on the projected implicit gradient descent algorithm to find a Stackelberg equilibrium of the transformed $1-1$ Stackelberg game. We validate the proposed approaches using the two problems of deriving operating strategies for EV charging stations: (1) the first problem is optimizing the one-time charging price for EV users, in which a platform operator determines the price of electricity and EV users determine the optimal amount of charging for their satisfaction; and (2) the second problem is to determine the spatially varying charging price to optimally balance the demand and supply over every charging station.