Discrete-time stochastic Stackelberg dynamic games with a large number of followers

We consider a class of discrete-time stochastic Stackelberg dynamic games with one leader and the N followers where N is sufficiently large. The leader and the followers are coupled through a mean field term, representing the average behavior of the followers. We characterize a Nash equilibrium at the followers level, and a Stackelberg equilibrium between the leader and the followers group. To circumvent the difficulty that arises in characterizing a Stackelberg-Nash solution due to the presence of a large number of followers, our approach is to imbed the original game in a class of mean-field stochastic dynamic games, where each follower solves a generic stochastic control problem with an approximated mean-field behavior and with an arbitrary control for the leader. We first show that this solution constitutes an -Nash equilibrium for the followers, where can be picked arbitrarily close to zero when N is large. We then turn to the leader's problem, and show that the associated local optimal control problem, constructed via the mean field approximation, admits an (₁; ₂)-Stackelberg equilibrium, where both ₁ and ₂ are arbitrarily close to zero as N becomes arbitrarily large. Numerical examples included in the paper illustrate the theoretical results.