Energy as a Concealable State in Adversarial UAV Patrolling: Formulation, an Energy-Security Threshold, and the Limits of Self-Play
We study energy-constrained adversarial patrolling on a graph, in which a battery-limited UAV defends a cluster of high-value targets against a strategic attacker who chooses when and where to strike. Unlike prior adversarial patrolling, the patroller must periodically return to a base to recharge; unlike prior energy-aware patrolling, it faces a self-interested adversary. Our central observation is that the remaining energy is a hidden state: the attacker never observes the battery directly, but observes the patroller's trajectory and can infer when a recharge excursion, and thus a vulnerability window, is imminent. We formalize the interaction as a zero-sum partially observable stochastic game and report a negative result on the solver side: neither independent deep Q-learning nor Neural Fictitious Self-Play reaches a stable equilibrium at this scale; each improves transiently and then collapses, with the co-trained thwart rate falling from about 0.25 to about 0.09 over training. Using a structural analysis independent of the learning dynamics, we show that achievable security rises monotonically with the energy budget, from zero below a threshold to about 0.7 when the budget is ample, establishing the energy budget as the primary determinant of defensibility. We set out the program the model is built to answer: whether an inference-capable attacker concentrates its successful strikes in the recharge window, and whether the defender can learn deceptive recharge timing to keep that window closed.