Pricing in Linear-Quadratic Dynamic Games

D. Calderone, S. Coogan, L. J. Ratliff, C. Tomlin, and S. S. Sastry
Fiftieth Annual Allerton Conference on Communication, Control, and Computing, 2012

Abstract

We investigate the use of pricing mechanisms as a means to achieve a desired feedback control strategy among selfish agents. We study a Stackelberg 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 to achieve consensus in a general network.