An Optimization and Control Theoretic Approach to Noncooperative Game Design

Abstract: This paper investigates design of noncooperative games from an optimization and control theoretic perspective. Pricing mechanisms are used as a design tool to ensure that the Nash equilibrium of a fairly general class of noncooperative games satisfies certain global objectives such as welfare maximization or achieving a certain level of quality-of-service (QoS). The class of games considered provide a theoretical basis for decentralized resource allocation and control problems including network congestion control, wireless uplink power control, and optical power control. The game design problem is analyzed under different knowledge assumptions (full versus limited information) and design objectives (QoS versus utility maximization) for separable and non-separable utility functions. The ``price of anarchy'' is shown not to be an inherent feature of full-information games that incorporate pricing mechanisms. Moreover, a simple linear pricing is shown to be sufficient for design of Nash equilibrium according to a chosen global objective for a fairly general class of games. Stability properties of the game and pricing dynamics are studied under the assumption of time-scale separation and in two separate time-scales. Thus, sufficient conditions are derived, which allow the designer to place the Nash equilibrium solution or to guide the system trajectory to a desired region or point. The obtained results are illustrated with a number of examples.