  We consider a Gaussian interference channel with independent direct and cross
link channel gains, each of which is independent and identically distributed
across time. Each transmitter-receiver user pair aims to maximize its long-term
average transmission rate subject to an average power constraint. We formulate
a stochastic game for this system in three different scenarios. First, we
assume that each user knows all direct and cross link channel gains. Later, we
assume that each user knows channel gains of only the links that are incident
on its receiver. Lastly, we assume that each user knows only its own direct
link channel gain. In all cases, we formulate the problem of finding a Nash
equilibrium (NE) as a variational inequality (VI) problem. We present a novel
heuristic for solving a VI. We use this heuristic to solve for a NE of power
allocation games with partial information. We also present a lower bound on the
utility for each user at any NE in the case of the games with partial
information. We obtain this lower bound using a water-filling like power
allocation that requires only knowledge of the distribution of a user's own
channel gains and average power constraints of all the users. We also provide a
distributed algorithm to compute Pareto optimal solutions for the proposed
games. Finally, we use Bayesian learning to obtain an algorithm that converges
to an $\epsilon$-Nash equilibrium for the incomplete information game with
direct link channel gain knowledge only without requiring the knowledge of the
power policies of the other users.
