In this paper, we derive and evaluate private objective functions for large-scale, distributed opportunistic spectrum access (OSA) systems. By means of any learning algorithms, these derived objective functions enable OSA users to assess, locate, and exploit unused spectrum opportunities effectively by maximizing the users' average received rewards. We consider the elastic traffic model, suitable for elastic applications such as file transfer and web browsing, and in which an SU's received reward increases proportionally to the amount of received service when the amount is higher than a certain threshold. But when this amount is below the threshold, the reward decreases exponentially with the amount of received service. In this model, SUs are assumed to be treated fairly in that the SUs using the same band will roughly receive an equal share of the total amount of service offered by the band. We show that the proposed objective functions are: near-optimal, as they achieve high performances in terms of average received rewards; highly scalable, as they perform well for small- as well as large-scale systems; highly learnable, as they reach up near-optimal values very quickly; and distributive, as they require information sharing only among OSA users belonging to the same band.