The invariant Nash equilibrium for stochastic games in multiple access channel

In this paper, we present an iterative best-response algorithm to compute the Nash equilibrium of a power allocation game in a multiple access channel (MAC), where each user greedily chooses a power allocation policy to maximize their average transmission rate. We consider a MAC where the fading channel gains are assumed to be stationary and ergodic processes, taking values from a finite set. The receiver decodes the message of a user by assuming the messages of the rest of the users as noise. The Shannon capacity of a user is considered to be the achievable rate of that user. A user transmits with a transmit power chosen from a finite set of power values, in a selfish manner, such that their average rate of transmission is maximized. We show that the Nash equilibrium of this game is unique, provided the number of users in the system is sufficiently large, but finite. Under this condition, we also show that the equilibrium policy does not change with more number of users coming into the system. We propose a simple greedy algorithm to compute the Nash equilibrium when the number of users is sufficiently large, but finite. The proposed algorithm does not depend upon the parameters of other users and hence, can be computed without any feedback or side-information from other users. We also present numerical results to illustrate the performance of the proposed algorithm.