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 $ε$-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.