Rate-Based Equilibria in Collision Channels with Fading

We consider a wireless collision channel, shared by a finite number of users who transmit to a common base station. Each user wishes to minimize its average transmission rate (or power investment), subject to minimum throughput demand. The channel quality between each user and the base station is randomly time-varying, and partially observed by the user through Channel State Information (CSI) signals. Assuming that all users employ stationary, CSI-dependent transmission policies, we investigate the properties of the Nash equilibrium of the resulting game between users. We characterize the feasible region of user's throughput demands, and provide lower bounds on the channel capacity that hold both for symmetric and non-symmetric users. Our equilibrium analysis reveals that, when the throughput demands are feasible, there exist exactly two Nash equilibrium points, with one strictly better than the other (in terms of power investment) for each user. We further demonstrate that the performance gap between the two equilibria may be arbitrarily large. This motivates the need for distributed mechanisms that lead to the better equilibrium. To that end, we suggest a simple greedy (best-response) mechanism, and prove convergence to the better equilibrium. Some important stability properties of this mechanism in face of changing user population are derived as well.