Downlink Power Control in Two-Tier Cellular OFDMA Networks Under Uncertainties: A Robust Stackelberg Game

We consider the problem of robust downlink power control in orthogonal frequency-division multiple access (OFDMA)-based heterogeneous wireless networks (HetNets) composed of macrocells and underlaying small cells. A non-cooperative setting is assumed where the macro base stations (MBSs) and small cell base stations (SBSs) compete with each other to maximize their own capacities considering imperfect channel state information. A robust Stackelberg game (RSG) is formulated to model this hierarchical competition where the MBSs and SBSs act as the leaders and the followers, respectively. The formulated RSG can be expressed as an equilibrium program with equilibrium constraints (EPEC). A comprehensive study of this RSG is provided considering various power constraints (e.g., total and spectral mask), various interference constraints (e.g., individual and global), and different uncertainty models (e.g., column-wise and ellipsoidal). We show how the different constraints and uncertainty models change the property of the game (e.g., Nash equilibrium problem (NEP) or generalized Nash equilibrium problem (GNEP)) and accordingly impact the choice of analysis method (e.g., game theory or variational inequality (VI)), solution (e.g., closed-form or numerical), and the design of algorithms and their distributive properties (e.g., totally distributed, semi-distributed, and centralized). A robust Stackelberg equilibrium (RSE) is considered to be the solution and its existence and uniqueness are investigated. Also, algorithms are proposed to arrive at the RSE. Numerical results show the effectiveness of robust solutions in an imperfect information environment.