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

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.