We consider heterogeneous networks with multiple femtocells and macrocells. Femto-base stations (femto-BS) are constrained to allocate transmitting powers such that the total interference at each macro-user terminal (macro-UT) is below a given threshold. We formulate a power allocation problem as a concave game with femto-BSs as players and multiple macro-UTs enforcing coupled constraints. Equilibrium selection is based on the concept of normalized Nash equilibrium (NNE). When the interference at a femto-user terminal (femto-UT) from adjacent femto-BSs is negligible, for any strictly concave nondecreasing utility the NNE is unique and the NNE is the solution of a concave potential game. We also propose a distributed algorithm which converges to the unique NNE. When the interference is not negligible, an NNE may not be unique and the computation of NNE has exponential complexity. 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 concept for the relevant case of Shannon capacity as femto-BS's utility.
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