Nowadays, ubiquitous network access has become a reality thanks to Unmanned Aerial Vehicles (UAVs) that have gained extreme popularity due to their flexible deployment and higher chance of Line-of-Sight (LoS) links to ground users. Telecommunication service providers deploy UAVs to provide flying network access in remote rural areas, disaster-affected areas or massive-attended events (sport venues, festivals, etc.) where full set-up to provide temporary wireless coverage would be very expensive. Of course, a UAV is battery-powered which means limited energy budget for both mobility aspect and communication aspect. An efficient solution is to allow UAVs swhiching their radio modules to sleep mode in order to extend battery lifetime. This results in temporary unavailability of communication feature. Within such a situation, the ultimate deal for a UAV operator is to provide a cost effective service with acceptable availability. This would allow to meet some target Quality of Service (QoS) while having a good market share granting satisfactory benefits. In this article, we exhibit a new framework with many interesting insights on how to jointly define the availability and the access cost in UAVempowered flying access networks to opportunistically cover a target geographical area. Yet, we construct a duopoly model to capture the adversarial behavior of service providers in terms of their pricing policies and their respective availability probabilities. Optimal periodic beaconing (small messages advertising existence of a UAV) is a vital issue that needs to be addressed, given the UAVs limited battery capacity and their recharging constraints. A full analysis of the game outcome, both in terms of equilibrium pricing and equilibrium availability, is derived. We show that the availability-pricing game exhibits some nice features as it is sub-modular with respect to the availability policy, whereas it is super-modular with respect to the service fee. Furthermore, we implement a learning scheme using best-response dynamics that allows operators to learn their joint pricing-availability strategies in a fast, accurate and distributed fashion. Extensive simulations show convergence of the proposed scheme to the joint pricing-availability equilibrium and offer promising insights on how the game parameters should be chosen to efficiently control the duopoly game.