Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions

We consider uplink power control for lognormal fading channels in the large population case. First, we examine the structure of the control law in a centralized stochastic optimal control setup. We analyze the effect of large populations on the individual control inputs. Next, we split the centralized cost to approach the problem in a game theoretic framework. In this context, we introduce an auxiliary LQG control system and analyze the resulting ε-Nash equilibrium for the control law; subsequently we generalize the methodology developed for the LQG problem to the wireless power control problem to get an approximation for the collective effect of all other users on a given user. The obtained state aggregation technique leads to highly localized control configurations in contrast to the full state based optimal control strategy.