Proposed algorithm. Here we elaborate the proposed bargaining algorithm that enables the transmitters to distributively design their beamforming vectors. We assume that there exists a feedback link from every receiver to every transmitter. The receivers use these links to feedback CSI. Each transmitter has CSI only on the links it is affecting. The transmitters are assumed synchronized, but no information (CSI or user data) is exchanged between them. In the algorithm, we use as bargaining value an upper bound on the interference generated by system i to system j. This bound, denoted cij, is adjusted in every iteration. During the bargaining, the receivers feed back a one-bit message that tells the transmitters whether the iteration was successful or not, i.e. whether the rates increased or not. We denote l the iteration counter, which also acts as a quantitative measure of the overhead (total number of bits per RX-TX feedback link) and the computational complexity (total number of optimization problems that need to be solved). A flowchart of the algorithm is illustrated in Figure 4.1. The first step of the algorithm is the decision whether the initialization point will be the NE or the ZF point. For this reason, the transmitters send two pilots using their MR and ZF beamforming vectors. The receivers measure the SINR for each transmission and feed back one-bit of information telling the transmitters which strategy yields higher SINR, hence rate. If Ri(wZF, wZF) ≥ Ri(wMR, wMR) for both systems, the algorithm is initialized with the ZF point, since it is closer than the NE to the Pareto boundary. Hence, the algorithm will require fewer iterations to converge to a solution. If only one system achieves higher rate with the ZF strategies, there is no incentive for the other to accept the ZF point as initial point. The algorithm is then initialized with the NE point. Then, the algorithm sets the stepsize for updating cij. As with any iterative algo- rithm, the best output is obtained for an infinitesimal stepsize. However, this is not practical, so we consider instead a fixed stepsize4. We assume that TXi samples the interval [0, pij(wMR)] uniformly in N + 1 points, to allow up to N iterations. This gives the step δij = ±pij(wMR)/N . The sign of δij depends on the initial point. If the algorithm is initialized with the NE, δij will be negative (decreasing interfer- ence). Otherwise, δij will be positive (increasing interference). At iteration l, TXi updates the interference level a...
