Deterministic and Stochastic Wireless Network Games: Equilibrium, Dynamics, and Price of Anarchy

Power control over wireless networks has been an active area of research with significant applied impact. A well-motivated line of this research, which has received increasing attention, is applying game-theoretic tools for both gaining insight and design of algorithms. In this paper, we build on the existing work and present a simple game-theoretic formulation of power control on wireless networks that incorporates two novel features. First, we do not impose exogenous power bounds on the feasible transmission power. Second, we allow the channel environment to be stochastic and time varying. Within this model, we first examine the deterministic game under a fixed environment, in which we develop a novel fixed-point theorem of independent interest that operates in general and unbounded partially ordered sets. We then leverage this customized fixed-point theorem to establish various equilibrium-related results: existence, uniqueness, and convergence, followed by a novel Price-of-Anarchy bound characterization. Finally, we study the stochastic behavior of the best response dynamics and establish a number of desirable properties in the presence of a stochastic and time-varying channel.