Normalized nash equilibrium for power allocation in cognitive radio networks

We consider a cognitive radio system consisting of several secondary networks and primary user-terminals (primary-UTs). In a secondary network a secondary-base station (secondary-BS) transmits to a secondary-user terminal (secondary-UT) with certain power. Secondary-BSs are constrained to allocate transmitting powers such that the total interference at each primary-UT is below a given threshold. We formulate the power allocation problem as a concave non cooperative game with secondary-BSs as players and multiple primary-UTs enforcing coupled constraints. The equilibrium selection is based on the concept of normalized Nash equilibrium (NNE). When the interference at a secondary-UT from adjacent secondary-BSs is negligible, the NNE is shown to be unique for any strictly concave utility. The NNE is also shown to be the solution of a concave potential game. We propose a distributed algorithm which converges to the unique NNE. When the interference at a secondary-UT from adjacent secondary- BSs is not negligible, an NNE may not be unique and the computation of the NNE has exponential complexity. To avoid these drawbacks, we introduce the concept of weakly normalized Nash equilibrium (WNNE) which keeps the most of NNEs' interesting properties but, in contrast to the latter, the WNNE can be determined with low complexity. We show the usefulness of the WNNE when the utility function is the Shannon capacity.