Nash Bargaining in Beamforming Games with Quantized CSI in Two-User Interference Channels

In this paper, we consider a beamforming game of the transmitters in a two-user multiple-input single- output interference channel using limited feedback and investigate how each transmitter should find a strategy from the quantized channel state information (CSI). In the beamforming game, each transmitter (a player) tries to maximize the achievable rate (a payoff function) via a proper beamforming strategy. In our case, each transmitter´s beamforming strategy is represented by a linear combining factor between the maximum-ratio transmission (MRT) and the zero-forcing (ZF) beamforming vectors, which is shown to be a Pareto optimal achieving strategy. With the perfect CSI, each transmitter can know the exact achievable rate region, and hence can find the beamforming strategy corresponding to any point in the achievable rate region. With limited feedback, however, the transmitters can only conjecture the achievable rate region from the quantized CSI, so their optimal strategies may not be optimal anymore. Considering the quantized CSI at the transmitter, we first find the Nash equilibrium in a non-cooperative game. Then, in a cooperative (Nash bargaining) game, we find a Nash bargaining solution and test its validity. Finally, we propose three bargaining solutions that improve the validity of the cooperation or the average Nash product. Our proposed bargaining solutions utilize the codebook structure; instead of each quantized channel itself, its Voronoi region is considered^.