Joint Sensing and Power Allocation in Nonconvex Cognitive Radio Games: Quasi-Nash Equilibria

In this paper, we propose a novel class of Nash problems for Cognitive Radio (CR) networks composed of multiple primary users (PUs) and secondary users (SUs) wherein each SU (player) competes against the others to maximize his own opportunistic throughput by choosing jointly the sensing duration, the detection thresholds, and the vector power allocation over a multichannel link. In addition to power budget constraints, several (deterministic or probabilistic) interference constraints can be accommodated in the proposed general formulation, such as constraints on the maximum individual/aggregate (probabilistic) interference tolerable from the PUs. To keep the optimization as decentralized as possible, global interference constraints, when present, are imposed via pricing; the prices are thus additional variables to be optimized. The resulting players' optimization problems are nonconvex and there are price clearance conditions associated with the nonconvex global interference constraints to be satisfied by the equilibria of the game, which make the analysis of the proposed game a challenging task; none of classical results in the game theory literature can be successfully applied. To deal with the nonconvexity of the game, we introduce a relaxed equilibrium concept - the Quasi-Nash Equilibrium (QNE)- and study its main properties, performance, and connection with local Nash equilibria. Quite interestingly, the proposed game theoretical formulations yield a considerable performance improvement with respect to current centralized and decentralized designs of CR systems, which validates the concept of QNE.