Linear-quadratic stochastic differential Stackelberg games with a high population of followers

Jun Moon, Tamer Basar

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We consider a class of stochastic differential games with the Stackelberg mode of play, with one leader and N uniform followers (where N is sufficiently large), where each player has its own local controlled dynamics and quadratic cost function, with the coupling between the players being through the cost functions. Particularly, the leader's cost function has as input the average value of the states of the followers, and each follower's cost function has a similar term in addition to being directly affected by the control function of the leader; thus, the leader controls the behavior of the followers (who play a Nash game) through his control strategy. As such, this class of stochastic differential games is quite difficult to analyze and obtain the Stackelberg-Nash solution of. To circumvent this difficulty, our approach in this paper is to imbed the original game in a class of mean-field stochastic differential games, where the followers solve individual stochastic control problems given the mean field behavior of their average states and with leader's control taken as an exogenous stochastic process. We show that for each fixed policy of the leader, the followers' optimal decentralized local policies lead to an ?-Nash equilibrium, where ? = O(1/√N). The paper then solves the leader's optimal control problem, as a constrained optimization problem, with the constraint being induced by the ?-Nash equilibrium policies of the followers (which depend on the leaders control as an exogenous process). We obtain the leaders optimal decentralized local control, which we subsequently show to constitute an O(1/√N)-approximate Stackelberg equilibrium for the original game. A numerical example included in the paper illustrates the theoretical results.

Original languageEnglish (US)
Title of host publication54rd IEEE Conference on Decision and Control,CDC 2015
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages2270-2275
Number of pages6
ISBN (Electronic)9781479978861
DOIs
StatePublished - Feb 8 2015
Event54th IEEE Conference on Decision and Control, CDC 2015 - Osaka, Japan
Duration: Dec 15 2015Dec 18 2015

Publication series

NameProceedings of the IEEE Conference on Decision and Control
Volume54rd IEEE Conference on Decision and Control,CDC 2015
ISSN (Print)0743-1546

Other

Other54th IEEE Conference on Decision and Control, CDC 2015
CountryJapan
CityOsaka
Period12/15/1512/18/15

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

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