This paper will discuss a series of game-based adversarial jamming beamforming problems in the electronic countermeasures domain, and aims to develop a unified method to solve these problems as well as finding their shared solution. In that scenario, a smart multi-antenna jammer attempts to minimize the maximum signal-to-interference-plus-jamming-plus-noise ratio (SIJNR) of multiple single-antenna users, while a smart multi-antenna base station tries to maximize the minimum SIJNR. According to the transmission order of jammer and communication system, the jamming problem can be modeled by the Nash game, the jammer-leader Stackelberg game, or the jammer-follower Stackelberg game. However, the jammer may find it enormously difficult to detect the transmission order, so it can not decide which game to choose. To address this issue, this paper will show that the three games share an equilibrium by introducing three surrogate problems and exploring their relations, so that the information of transmission order is redundant. Subsequently, to find the shared equilibrium, the surrogate problems are transformed into the problems of the mathematical program with equilibrium constraint, and a convex-concave-procedure-based penalty method is developed to solve them. The numerical results verify the existence of the shared equilibrium and demonstrate the superior performance of the proposed method.

Recently, various wireless Internet-of-Things devices and unmanned aerial vehicles have been frequently used to facilitate daily life. On the other hand, they can also pose serious threats to public security if maliciously used. A countermeasure against these threats is to transmit jamming signals to break the communication links between attacker and wireless devices. However, this is not easy since modern multi-device (multi-user) wireless communication systems are smart in avoiding jamming. Moreover, a relay is widely used to improve the jamming resistance. To successfully jam those malicious devices, we consider a jamming and anti-jamming zero-sum game, in which the attacker tries to maximize the sum-rate of a relay-assisted wireless system and the jammer tries to minimize it. Finding the equilibrium is challenging due to its non-convexity and complex structure. We address this problem in two cases. Specifically, in the single-device (single-user) case, we show that putting the whole jamming power to either relay or device achieves the Nash equilibrium. In the multi-device case, an efficient method is developed, which transforms this game into a min-max optimization problem and decouples it into two sub-problems by the hybrid block successive approximation method. The resultant sub-problems are solved by the multi-block alternating direction method of multipliers and the block successive upper-bound minimization method of multipliers, respectively. Then, a stationary point of the original problem can be iteratively achieved. Numerical results demonstrate that the two proposed jamming methods outperform many traditional methods.