# Mean-Field Stackelberg Game-Based Security Defense and Resource Optimization in Edge Computing  
[PERSON]  
1 School of Information Engineering, Ningxia University, Yinchuan 750021, China; [EMAIL_ADDRESS] (S.L. ); [EMAIL_ADDRESS] (X.W)1 School of Information Engineering, Ningxia University, Yinchuan 750021, China; [EMAIL_ADDRESS] (S.L. ); [EMAIL_ADDRESS] (X.W)1 School of Information Engineering, Ningxia University, Yinchuan 750021, China; [EMAIL_ADDRESS] (S.L. ); [EMAIL_ADDRESS] (X.W)1 School of Information Engineering, Ningxia University, Yinchuan 750021, China; [EMAIL_ADDRESS] (S.L. ); [EMAIL_ADDRESS] (X.W)1 School of Information Engineering, Ningxia University, Yinchuan 750021, China; [EMAIL_ADDRESS] (X.W)1 School of Information Engineering, Ningxia University, Yinchuan 750021, China; lis@mxu. This study concentrates on addressing security defense challenges in edge computing environments to identify and implement optimal defense strategies. We examine the interaction behavior between defenders and attackers by using the mean-field Stackelberg game theory [10; 11; 12], which can solve complex and dynamic problems with large players. In the mean-field game model, the interaction behavior of individuals can be coupled through the mean-field term, and then the global problem can be converted into individual subproblems that greatly reduce the computing complexity of large-scale networks. The mean-field game (MFG) model has been applied to the security defense problem in [13; 14]. For large-scale edge devices, the authors in [13] designed an anti-attack model based on the mean-field game and obtained the equilibrium through a self-organizing neural network. In [14], the authors proposed a finite-horizon indefinite mean-field stochastic cooperative linear-quadratic difference game and analyzed the balance between the minimization of investments and the security level. In this paper, we consider the number of attackers as one player, which is modeled as the leader, while the defenders are the followers. In this case, the leader first chooses and then announces the optimal strategy to the defenders. Each defender will choose its optimal defense strategy to minimize the loss based on the leader's observed strategy. We aim to obtain the optimal defense strategies based on the minimization of resource consumption and the strategy of the attacker. Meanwhile, the objective is to balance the profits and the resource consumption for both the defenders and the attackers. The contributions of this article can be summarized as follows. 1. Firstly, we analyze an edge computing system environment where the attacker is the leader, while the defenders are the followers. We propose an optimization problem that jointly optimizes resource consumption and player decisions by including state and decision variables. 2. Secondly, we formulate a mean-field Stackelberg game model to analyze the optimization problem, in which the dynamic evolutions of the states of the defenders are coupled with each other through the mean-field term and strongly influenced by the attack intensity. Moreover, we analyze the impact of the defense strategy on the evolution of the state of the attacker. The objectives for the defenders are to minimize the cost of defending against attackers and reduce the losses caused by attacks. The objective for the attackers is to minimize their attack cost. 3. Finally, we solve a local optimal control problem of the defenders given an arbitrary strategy of the attackers and discuss how the defenders' optimal decentralized strategies lead to an \(\varepsilon\)-Nash equilibrium for each fixed strategy of the leader, where \(\varepsilon\) converges to zero as \(N\to\infty\). We then consider the leader's local optimal control problem and obtain the leader's decentralized optimal controller. The remainder of this article is organized as follows. The related works are introduced in Section 2.