Mean-field games and green power control

In this work, we consider a distributed wireless network where many transmitters communicate with a common receiver. Having the choice of their power control policy, transmitters are concerned with energy constraints: instantaneous energy-efficiency and long-term energy consumption. The individual optimization of the average energy-efficient utility over a finite horizon is studied by using control theory and a coupled system of Hamilton-Jacobi-Bellman-Fleming equations is obtained. Even though the existence of a solution to the corresponding stochastic differential game is proven, the game is difficult to analyze when the number of transmitters is large (in particular, the Nash equilibrium analysis becomes hard and even impossible). But when the number of transmitters is large, the stochastic differential game converges to a mean-field game which is ruled by a more tractable system of equations. A condition for the uniqueness of the equilibrium of the mean-field game is given.