Spectrum sharing between two independent, co-existing transmit-receive pairs (TRPs) is formulated as a non-cooperative game with the TRPs as players, their individual link rates as payoffs and power allocation over the utilized spectral bands as the strategy. A Nash Equilibrium (NE) corresponds to the outcome of such a game and TRPs iteratively use the water-filling algorithm according to an agreed order for achieving the NE. Dynamics of this distributed algorithm is studied to determine the conditions for convergence and characterize the resulting NE. A sufficient condition on global convergence is derived and is shown to be tighter than existing ones. Further, a novel characterization of the globally achievable NE based on necessary conditions is presented. Some of these results are also extended to multiple NE scenarios where local convergence is exhibited.