This paper describes an adaptation of cognitive radio technology for tactical wireless networking. We introduce Competing Cognitive Radio Network (CCRN) featuring both communicator and jamming cognitive radio nodes that strategize in taking actions on an open spectrum under the presence of adversarial threats. We present the problem in the Multi-armed Bandit (MAB) framework and develop the optimal media access strategy consisting of mixed communicator and jammer actions in a Bayesian setting for Thompson sampling based on extreme value theory. Empirical results are promising that the proposed strategy seems to outperform Lai & Robbins and UCB, some of the most important MAB algorithms known to date.