In this paper we investigate the impact that incomplete knowledge regarding user activity can have on the equilibrium transmission strategy for an OFDM-based communication system. The problem is formulated as a two user non-zero sum game for independent fading channel gains, where the equilibrium strategies are derived in closed form. This allows one to show that a decrease in uncertainty about the user activity could reduce the number of subcarriers jointly used by the users. For the boundary case (with complete information, which reflects a classical water-filling game) the equilibrium strategies are given explicitly. The necessary and sufficient conditions, when channels sharing strategies are optimal, is established as well as the set of shared subcarriers is identified. The stability of the upper bound of the size of this set with respect to power budgets is derived.