Unmanned aerial vehicles (UAVs) were initially developed for military monitoring and surveillance tasks but found several interesting applications in the civilian domain. A promising application/technology is to use drone small cells (DSCs) to expand wireless communication coverage on demand. Rapid deployment along with limited operating costs are key factors that boost the development of DSCs for both military and civilian utilizations. DSCs are rapidly deployable to provide connectivity for temporary users (e.g. attendees of festivals, sporting events, or seminars), or over disaster areas to replace damaged communication infrastructure. UAVs are battery-powered, which makes energy consumption optimization a critical issue for acceptable performance, high availability, and an economically viable DCS deployment. In this article we focus on the scheduling of beaconing periods as an efficient means of energy consumption optimization. The conducted study provides a sub-modular game perspective of the problem and investigates its structural properties. We also provide a learning algorithm that ensures convergence of the considered UAV network with its unique Nash equilibrium operating point. Finally, we conduct extensive numerical investigations to assist our claims about the energy efficiency of the strategic beaconing policy (at Nash equilibrium).