Invariant Nash Equilibrium for Large Player Games in Multiple Access Channels

We consider a wireless multiple access channel (MAC) with N users. Associated with each user is their time-varying channel state and a finite-length queue which varies with time. In MAC, a receiver decodes the signals of each user by treating the other users' signals as noise. Each user decides their transmit power and the queue-admission control variable dynamically to maximize their expected throughput without any knowledge of the states and actions of other users. This decision problem is formulated as a Markov game for which we show the existence of equilibrium and an algorithm to compute the equilibrium policies. We show that, when the number of users exceeds a given threshold, the expected throughput of all users at all the equilibria points are the same. Furthermore, we show that the equilibrium policies of the users are invariant as long as the number of users remain above the latter threshold, which is referred to as the infinitely invariant Nash equilibrium (IINE). For the considered system, we prove that IINE exists and show that each user can compute these policies using a sequence of linear programs which does not depend upon the parameters of the other users. We also provide the necessary and sufficient conditions for the existence of IINE. Finally, we validate our analysis using numerical simulations.