Pricing in linear-quadratic dynamic games

We investigate the use of pricing mechanisms as a means to achieve a desired feedback control strategy among selfish agents. We study a hierarchical linear-quadratic game with many dynamically coupled Nash followers and an uncoupled leader. The leader influences the game by choosing the quadratic dependence on control actions for each follower´s cost function. We show that determining whether the leader can establish the desired feedback control as a Nash equilibrium among the followers is a convex feasibility problem for the continuous-time infinite horizon, discrete-time infinite horizon, and discrete-time finite horizon settings, and we present several extensions to this main result. In particular, we discuss methods for ensuring that the total cost incurred due to the leader´s pricing is as close as possible to a specified nominal cost, as well as methods for minimizing the explicit dependence of a player´s cost on other players´ control inputs. Finally, we apply the proposed method to the problem of ensuring the security of a multi-network and to the problem of pricing for controlled diffusion in a general network.