We consider the decentralized power optimization problem for Gaussian fast-fading multiple access channel (MAC) so that the average sum throughput is maximized. In our MAC setup, each transmitter has access to only its own fading coefficient or channel state information (CSI), while the receiver has full CSI available at all instants. Unlike centralized MAC (full CSIT MAC) where the optimal powers are known explicitly, the analytical solution for optimal decentralized powers does not seem feasible. In this letter, we specialize the alternating-maximization (AM) method for numerically computing the optimal powers and ergodic capacity of the decentralized MAC for general fading statistics and average power constraints. For illustration, we apply our AM method to compute the capacity of MAC channels with fading distributions such as Rayleigh and Rician.