Cheat-Proof Distributed Power Control in Full-Duplex Small Cell Networks: A Repeated Game With Imperfect Public Monitoring

We address the problem of distributed power control in a two-tier cellular network, where full-duplex small cells underlay a macro cell in a co-channel deployment scenario. We first formulate the distributed power control problem as a non-cooperative game and then extend it to a repeated game with imperfect public monitoring. The repeated game formulation prevents deceitful small cells from deviating from the social optimal solution for their own benefit. We establish the existence and uniqueness of the Nash equilibrium in the formulated non-cooperative game. We also characterize the set of public perfect equilibrium for the repeated game. A two-phase distributed algorithm is proposed to achieve and enforce a Pareto optimal transmit power profile. The solution obtained by this algorithm is also social optimal. Phase 1 of the algorithm is a fully distributed learning phase based on perturbed Markov chains, where each base station individually learns a Pareto optimal operating point. Phase 2 is composed of two rules: 1) a detection rule based on Page-Hinckley test to detect cheating and 2) a punishment rule to motivate cheating base stations to cooperate. Through theoretical analysis, we prove that the proposed distributed power control mechanism achieves a public perfect equilibrium point of the formulated repeated game. The power control algorithm is also cheat-proof and needs only a small amount of information exchange among network nodes. The effectiveness of the algorithm is shown through numerical analysis. Our proposed model, algorithm, and analysis are also valid for a half-duplex system as a special case.