Battery-state dependent power control as a dynamic game

Consider an uplink cellular network shared by a finite number of mobile users with limited batteries. Whenever the battery drains out, the user pays a fixed price to recharge the battery. Users, assumed to have always traffic to send, control their transmission power in a noncooperative way. The novelty of our model is in considering the dynamic game in which the transmission power of a player may depend on the amount of energy left in its battery. We consider various models and various types of constraints and derive for each one the structure of the equilibrium. A particular interesting structure is obtained when there are constraints on the maximum transmission power which become tighter as the battery drains out. Using Schur convexity and majorization, we identify an equilibrium where each mobile distributes the power of each battery along the batterypsilas lifetime in a way equivalent to a max-min assignment.