Game Theoretic Framework for Beamforming Optimization for Photon-Counting Multiuser MISO Channel with Asymptotic Low Power

Abstract

We propose new game theoretic frameworks that permit to maximize the performance of an interference-limited photon-counting multiuser Poisson channel with asymptotically low transmit power considered. For a memoryless stationary Poisson channel, we consider a conflicting situation where multiple users share the same spectrum and attempt to maximize their throughput. The system model is constructed, where multiple transmitters communicate with many users each equipped with a single photon-counting receiver. Importantly, to limit the interference, we consider the transmission at an asymptotically low-power regime. The first contribution of this paper is to model the problem as a non-cooperative game and present the Nash equilibrium beamformer. We present the sufficient and necessary conditions for the Nash equilibrium solution. Next, we develop a cooperative game, where users can autonomously bargain to maximize their achievable capacity. To obtain the Nash bargaining solution, we generate the convex hull by performing time sharing. The key idea utilized in generating a convex hull is to choose different strategies at different time slots. The bargaining among the users is shown to yield a significant performance improvement at asymptotically low transmit power. Moreover, it is shown that selfish but cooperative users can achieve maximum sum-rate performance.