# Research on the Optimization Management of Cloud Privacy Strategy Based on Evolution Game

Authors: Pan Jun Sun
Journal: Security and Communication Networks
 (2020)

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## Abstract

Cloud computing services have great convenience, but privacy security is a big obstacle of popularity. In the process result of privacy protection of cloud computing, it is difficult to choose the optimal strategy. In order to solve this problem, we propose a quantitative weight model of privacy information, use evolutionary game theory to establish a game model of attack protection, design the optimal protection strategy selection algorithm, and make the evolutionary stable equilibrium solution method from the limited rational constraint. In order to study the strategic dependence of the same game group, the classical dynamic replication equation is improved by using the incentive coefficient, an improved evolutionary game model of attack protection is constructed, the stability of equilibrium point is further analyzed by Jacobian matrix method, and the optimal selection strategy is obtained under different conditions. Finally, the correctness and validity of the model are verified by experiments, different strategies of the same group have the dual effects of promotion and inhibition, and the advantages of this paper are shown by comparing with other articles.


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## Body

## 1. Introduction

With the development of information technology, the scale of the network is becoming more and more complex, and the serious privacy leak events cause tremendous harm to cloud computing [1]. To protect privacy of network service, firewalls, intrusion detection, and antivirus software technologies have been widely used. However, invaders can only be detected after the event, which often causes serious losses.Advanced persistent threat attackers use a variety of complex methods to steal information continuously and secretly from the cloud storage system, such as spear phishing and waterhole attack. They can even induce the cloud storage system to apply specific defense strategies and attack them, which is a great threat to cloud computing. Therefore, it is necessary to research the new privacy protection technologies [2].An ideal protection system should protect all weak points or attacks. However, considering the limitation of organizational resources, we must consider the concept of “moderate security” to find a balance between information privacy security risk and investment and make the most reasonable decision with limited resources. Whether the strategy of protector is effective should depend not only on its own behavior, but also on the strategies of the attacker and the system. Therefore, the game theory can be used to study the information security problems such as the conflict of attack and protect and the optimal protection decision.The basic characteristics of game theory are the opposition of objectives; both the dependence of strategies and the noncooperation of relationships are in the process of privacy protection. In the process of the game, because of the learning mechanism, low-payoff participants constantly learn the strategies of high-payoff participants and improve their behaviors [3].Based on bounded rationality, the evolutionary game continuously improves the intrinsic driving force of behavior strategy through learning mechanism, which can effectively enhance the accuracy and reliability of the game model [3, 4]. This paper constructs an evolutionary game model of attack and protect and analyzes the rules of stability and equilibrium evolution. So, the main contributions of our paper are as follows:(i)
We propose a quantitative weight model of privacy information, construct an evolutionary game model of attack-protect, make an evolutionary stable strategy solution method, achieve the optimal privacy strategy selection, and analyze the different evolutionary stable equilibrium(ii)
To express the strategic dependence, we propose improving the accuracy of the replication dynamic equation by incentive coefficients, construct an improved evolutionary game model of cloud computing privacy protection, and give the detailed process of equilibrium solution(iii)
We use the Jacobian matrix method to analyze the stability of the game equilibrium point and the evolutionary trend of the game and obtain the optimal protect strategyThe structure of this paper is as follows. In Section2, the related research work is introduced. In Section 3, this paper constructs an evolutionary game model of attack and protect and analyzes the different evolutionary equilibrium. In Section 4, this paper proposes an improved evolutionary game model by incentive coefficients and analyzes the stability of the evolutionary game. In Section 5, this paper designs relevant experiments. In Section 6, this paper summarizes the work and future research directions. To understand this article, a framework is given in Figure 1.Figure 1
The architecture of the evaluation game model and simulation experiments.


## 2. Related Work

The main security privacy strategy technologies for cloud system can be classified into three categories: attack defense game, evolution game, and strategy selection.


### 2.1. Attack Defense Game Approach

Neupane et al. [5] proposed a new defense system based on the camouflage theory, which can prevent the attackers from the analysis of attack characteristics and can reduce the impact of the target attack on the high-value services hosted in the cloud platform by “isolating virtual machine” and strategy coordination. Xiao et al. [6, 7] described the interaction between defenders and attackers in the cloud storage system, studied a mixed strategy in storage defense game, and proved that the view of the attackers can improve the effectiveness of defenders. Li et al. [8] proposed an attack-defend game model, which was a two-person zero-sum static game with complete information. When the price parameter exceeded the threshold, the defender will switch to the protection mode. Lv et al. [9] proposed a dynamic defense model based on the mixed strategy game, which optimized the allocation of the limited security resources of the target network, and allocated the dynamic optimal defense strategy for each node at different times. Based on the game between the data owner and data requester, Sfar et al. [10] proposed a solution to protect privacy in the context, which used incentives for privacy concessions or active attacks to describe game elements and found a balance between privacy concessions and incentives. Min et al. [11] derived the Nash equilibrium of symmetric and asymmetric allocation game between the attacker and the defender, initialized the quality value by using the experience in similar scenarios, and obtained the optimal defense performance. Hota and Sundaram [12] studied the influence of node’s behavior probability weight and the distribution of security risk and described the graph topology of the average attack probability under the Nash equilibrium in the weakest game.Stackelberg games are very useful for decision making in attack and defense scenario. Xiao et al. [6] deduced the Nash equilibrium of detection game and proposed a detection scheme based on strategy hill-climbing, which increased the uncertainty of strategy to deceive attacker in dynamic game. So Jakóbik et al. [13] defined a Stackelberg game model, which allowed the automatic selection of provider level security decisions and maximized the benefits of defenders. Wahab et al. [14] proposed a repeated Bayesian Stackelberg game, which provided the optimal distribution of detection for virtual machines, increased the detection ability of attack, and greatly reduced the number of attacks.


### 2.2. Evolutionary Game Approach

Khalifa et al. [15] proposed a new tool to simulate the evolution of several populations, defined three different stable levels, strong, weak, and medium, which improved the accuracy and adaptability of the evolutionary game. Jiang et al. [16] studied the information reliability of a series of cooperative networks and proposed an evolutionary game model based on closed expression and reinforcement assistant method. Tan et al. [17] studied the strategy selection problem of evolutionary game dynamics with group interaction and obtained the cooperative conditions of public goods game and volunteer dilemma game. Based on the bounded rationality of the players, Hu et al. [18] established a game model of attack and defense under incomplete information, which extended the game strategy in the game structure. Du et al. [19] used the evolutionary game theory framework of community to analyze the privacy protection behavior of the social network and designed incentives based on cost performance. Based on the analytic hierarchy process, Zhang et al. [20] comprehensively analyzed the impact of mobile target defense technology, proposed an effective strategy selection algorithm based on joint defense, and selected many variation elements to defend different attack.


### 2.3. Game Strategy Selection Approach

Various game methods have been developed to study the privacy strategy selection between proctors and adversaries. Kamhoua et al. [21] studied the security of cloud computing participants and internal interdependence, analyzed many possible Nash equilibria, and described the adversary’s motivation more accurately. In order to study heterogeneous network system and provide the optimal security detection strategy, Wu et al. [22] gave the analysis of Bayesian equilibrium and robust Nash equilibrium with incomplete information and proposed a verification and calculation method for continuous kernels.Tan et al. [17] studied the strategy selection problem of evolutionary game dynamics with group interaction and obtained the cooperative conditions of public goods game and volunteer dilemma game. Cheng et al. [23] established a mobile target defense game model based on incomplete information and designed an optimal strategy algorithm to prevent the selection strategy from deviating from the actual network conditions.After synthetically analyzing the influence of defense cost and benefit on strategy selection, Cheng et al. [24] designed an optimal strategy selection algorithm, which corrected the deviation between the selected strategy and the actual network conditions, thus ensuring the correctness of the optimal strategy selection. Zhang et al. [25] emphasized the applicability and advantages of multiobjective in network security, proposed a multiobjective game model, and introduced and demonstrated the set strategy selection technology. Armstrong et al. [26] proposed a threat specific risk assessment method, which allowed administrators to make fine-grained decisions for the selection of mitigation strategies.


### 2.4. Discussion and Features of Our Research

In the attack defense game approach, the existing research is mainly to explore deterministic strategies, so we need to study the application of stochastic strategies in practice. In the evolutionary game behavior, the current research is mainly to calculate the equilibrium through the income matrix and focuses on the optimization of the calculation process, rarely summarizing the dynamic process of strategic evolution. In terms of application scenarios, both the behavior analysis of cloud computing networks and the dynamic research of privacy security are based on strategy selection, which makes the research on strategy selection more universal. The game is a continuous process, different strategies in the same group have a great impact on the game results, but there are few researches in these above articles.Our research is more general in network security protection, because the strategy set of our model can be extended toN, and both attackers and defenders can choose any strategy, which can be applied to general protection strategy choice. On the basis of bounded rationality, we consider and quantitatively describe the influence of similar strategies, introduce the influence factors, establish the evolutionary game model of attack and protect based on the improved dynamic replication equation, provide the detailed stability analysis, and improve the accuracy of the model.


## 2.1. Attack Defense Game Approach

Neupane et al. [5] proposed a new defense system based on the camouflage theory, which can prevent the attackers from the analysis of attack characteristics and can reduce the impact of the target attack on the high-value services hosted in the cloud platform by “isolating virtual machine” and strategy coordination. Xiao et al. [6, 7] described the interaction between defenders and attackers in the cloud storage system, studied a mixed strategy in storage defense game, and proved that the view of the attackers can improve the effectiveness of defenders. Li et al. [8] proposed an attack-defend game model, which was a two-person zero-sum static game with complete information. When the price parameter exceeded the threshold, the defender will switch to the protection mode. Lv et al. [9] proposed a dynamic defense model based on the mixed strategy game, which optimized the allocation of the limited security resources of the target network, and allocated the dynamic optimal defense strategy for each node at different times. Based on the game between the data owner and data requester, Sfar et al. [10] proposed a solution to protect privacy in the context, which used incentives for privacy concessions or active attacks to describe game elements and found a balance between privacy concessions and incentives. Min et al. [11] derived the Nash equilibrium of symmetric and asymmetric allocation game between the attacker and the defender, initialized the quality value by using the experience in similar scenarios, and obtained the optimal defense performance. Hota and Sundaram [12] studied the influence of node’s behavior probability weight and the distribution of security risk and described the graph topology of the average attack probability under the Nash equilibrium in the weakest game.Stackelberg games are very useful for decision making in attack and defense scenario. Xiao et al. [6] deduced the Nash equilibrium of detection game and proposed a detection scheme based on strategy hill-climbing, which increased the uncertainty of strategy to deceive attacker in dynamic game. So Jakóbik et al. [13] defined a Stackelberg game model, which allowed the automatic selection of provider level security decisions and maximized the benefits of defenders. Wahab et al. [14] proposed a repeated Bayesian Stackelberg game, which provided the optimal distribution of detection for virtual machines, increased the detection ability of attack, and greatly reduced the number of attacks.


## 2.2. Evolutionary Game Approach

Khalifa et al. [15] proposed a new tool to simulate the evolution of several populations, defined three different stable levels, strong, weak, and medium, which improved the accuracy and adaptability of the evolutionary game. Jiang et al. [16] studied the information reliability of a series of cooperative networks and proposed an evolutionary game model based on closed expression and reinforcement assistant method. Tan et al. [17] studied the strategy selection problem of evolutionary game dynamics with group interaction and obtained the cooperative conditions of public goods game and volunteer dilemma game. Based on the bounded rationality of the players, Hu et al. [18] established a game model of attack and defense under incomplete information, which extended the game strategy in the game structure. Du et al. [19] used the evolutionary game theory framework of community to analyze the privacy protection behavior of the social network and designed incentives based on cost performance. Based on the analytic hierarchy process, Zhang et al. [20] comprehensively analyzed the impact of mobile target defense technology, proposed an effective strategy selection algorithm based on joint defense, and selected many variation elements to defend different attack.


## 2.3. Game Strategy Selection Approach

Various game methods have been developed to study the privacy strategy selection between proctors and adversaries. Kamhoua et al. [21] studied the security of cloud computing participants and internal interdependence, analyzed many possible Nash equilibria, and described the adversary’s motivation more accurately. In order to study heterogeneous network system and provide the optimal security detection strategy, Wu et al. [22] gave the analysis of Bayesian equilibrium and robust Nash equilibrium with incomplete information and proposed a verification and calculation method for continuous kernels.Tan et al. [17] studied the strategy selection problem of evolutionary game dynamics with group interaction and obtained the cooperative conditions of public goods game and volunteer dilemma game. Cheng et al. [23] established a mobile target defense game model based on incomplete information and designed an optimal strategy algorithm to prevent the selection strategy from deviating from the actual network conditions.After synthetically analyzing the influence of defense cost and benefit on strategy selection, Cheng et al. [24] designed an optimal strategy selection algorithm, which corrected the deviation between the selected strategy and the actual network conditions, thus ensuring the correctness of the optimal strategy selection. Zhang et al. [25] emphasized the applicability and advantages of multiobjective in network security, proposed a multiobjective game model, and introduced and demonstrated the set strategy selection technology. Armstrong et al. [26] proposed a threat specific risk assessment method, which allowed administrators to make fine-grained decisions for the selection of mitigation strategies.


## 2.4. Discussion and Features of Our Research

In the attack defense game approach, the existing research is mainly to explore deterministic strategies, so we need to study the application of stochastic strategies in practice. In the evolutionary game behavior, the current research is mainly to calculate the equilibrium through the income matrix and focuses on the optimization of the calculation process, rarely summarizing the dynamic process of strategic evolution. In terms of application scenarios, both the behavior analysis of cloud computing networks and the dynamic research of privacy security are based on strategy selection, which makes the research on strategy selection more universal. The game is a continuous process, different strategies in the same group have a great impact on the game results, but there are few researches in these above articles.Our research is more general in network security protection, because the strategy set of our model can be extended toN, and both attackers and defenders can choose any strategy, which can be applied to general protection strategy choice. On the basis of bounded rationality, we consider and quantitatively describe the influence of similar strategies, introduce the influence factors, establish the evolutionary game model of attack and protect based on the improved dynamic replication equation, provide the detailed stability analysis, and improve the accuracy of the model.


## 3. A Game Model of Attack-Protect

In this section, we propose a quantitative weight model of privacy information, construct a game model of privacy protection, and analyze the evolution of the strategy selection mechanism to realize the optimal protect strategy. To read this article, some parameters are given in Table1.Table 1
Parameters and meanings.
ParametersMeaningsPIs=PI1,PI2,…,PInPrivacy information setG,S,T,UEvolutionary game modelGP and GAProtector and adversaryPS1,PS2,…,PSnStrategy set for protectorAS1,AS2,…,ASmStrategy set for adversaryT=p,qGame beliefsqiProbability of strategyPSipiProbability of strategyASiaijPayoff of adversary adoptASibijPayoff of protector adoptPSiUPExpected payoff of protectorUAExpected payoff of adversaryUP¯Average payoff of protectorUA¯An attack-protect payoff of adversaryxitNumber of players choosingPSiαiαi>0Protect strategy influence factoryitNumber of players choosingASiβiβi>0Attack strategy influence factorθijProtect strategy incentive coefficientΔijAttack strategy incentive coefficient


### 3.1. Privacy Metric Space

To make the expression of privacy information, this paper proposes a measurement model and proposes the distance between any elements in the metric space.Definition 1.
Suppose thatℜ is a nonempty set metric space; for two elements x, y in the ℜ, Φx,y represents the distance between two elements, which has two characters:(1)
Φx,y≥0, Φx,y=0, and x=y(2)
Ifz∈ℜ, Φx,y<Φx,z+Φy,z
Φx,y is the distance between two points x, y. ℜ,Φ is called as metric space according to the distance.
According to the definition of metric space, the following properties can be obtained:(1)Φx,y=Φy,x,x,y,z∈ℜ,Φx,z−Φy,z<Φx,y.
Norm is a basic concept in performance analysis, which is often used to measure the length or size of each element in the metric space.
Letnn1,n2,…,nk be a vector, and Η=hi,jm×n is a matrix.
Vector 1-norm is(2)n1=∑i=1kni.
Vector 2-norm is(3)n2=∑i=1kni21/2.
Matrix F-norm is(4)HF=∑i=1m∑j=1nhi,j21/2.
In this paper, the privacy vector is considered as an index in the measurement space, 2-norm is used to represent the size of the privacy value.
Assuming that a piece of privacy information isPIi=g1,g2,…,gi,…,gn, wi represents the privacy factor related to the user’s privacy; then the privacy value PIi of PIi can be expressed as(5)PIi=w1g12+w2g22+⋯+wngn22=∑i=1nwi×gi22,i=1,2,…,n.wi is the weight coefficient of the influence factor gi.
The correlation coefficient is an objective weight method to eliminate the influence of duplicate information on the comprehensive evaluation results. Calculate the correlation coefficient matrix; the original data containsn factors. Then the correlation coefficient matrix is(6)E=e11e12⋯e1ne21e22⋯e2n⋮⋮⋱⋮en1en2⋯enn.
After standardization, it can be simplified as(7)E=eijnn.
Calculate the sum value of1−rij in the jth column as follows:(8)∑1−ei1,∑1−ei2,…,∑1−eij.
The result of∑1−eij row vector is larger, the influence in the comprehensive evaluation system is greater, and the weight is more. Both privacy variability and conflict factor need to be considered in weight. In the paper, the objective weight method is adopted, and factor variability is represented by the standard deviation to show the difference between evaluation schemes of a factor. The larger the standard deviation is, the larger the value is.
Assume thatIi is the information quantity of ith factor; the various results of indicators can be considered by selecting the standard deviation σi. The conflict characteristics between the ith standard and other standards are measured by ∑1−eij. eij represents the correlation coefficient between the ith and the jth factors, and Ii can be expressed as follows:(9)Ii=σi∑i=1n1−eij.
The result ofIi is larger, the amount of data in the ith criterion is larger, and the importance is more. Therefore, the weight wi of the ith factor is as follows:(10)wi=Ii∑i=1′nIi.


### 3.2. Attack-Protect Evolutionary Game Model

In order to protect privacy informationPIs=PI1,PI2,…,PIn, we propose an attack-protect game model (Figure 2).Figure 2
Basic game tree of adversary-protector.Definition 2.
An attack-protect evolutionary game model can be defined as a 4-tupleAPEGM=G,S,T,U.
G=GP,GA is the participant space of evolutionary game, GP is the protector, and GA is the adversary.
S=PS,AS is game action strategy space, PS=PS1,PS2,…,PSn represents an optional strategy set for the protector, and AS=AS1,AS2,…,ASm represents an optional strategy set for the adversary. Both adversary and protector have multiple strategies.
T=pi,qi is a set of game beliefs, qi represents the probability set that the protector chooses a strategy PSi, and pi represents the probability set that the adversaries choose a strategy ASi.
U=UP,UA is a set of game payoff functions, UP represents the game payoffs of protectors, and UA expresses the game payoffs of adversaries.
In the game, both protector and adversary havePS=PS1,PS2,…,PSn and AS=AS1,AS2,…,ASm, respectively, m,n∈N, m,n≥2. When different strategies are used in the attack-protect game, both aij and bij represent the payoffs of adversary and protector when they adopt ASi and PSi. We can get the following formula:(11)a11,b11a12,b12⋯a1n,b1na21,b21a22,b22⋯a2n,b2n⋮⋮⋱⋮am1,bm1am2,bm1⋯amn,bmn.
In thePS=PS1,PS2,…,PSn, the player chooses PSi with qi, and q1+q2+⋯+qn=1. Similarly, in the AS=AS1,AS2,…,ASm, adversary selects ASi with pi, p1+p2+⋯+pm=1. Further, we can get the expected payoff UPSi and average payoff UP¯ of different protection strategies:(12)UPS1=p1b11+p2b21+⋯+pmbm1,UPS2=p1b12+p2b22+⋯+pmbm2,⋮UPSn=p1b1n+p2b2n+⋯+pmbmn,(13)UP¯=q1UPS1+q2UPS2+⋯+qnUPSn.
In strategy setPS=PS1,PS2,…,PSn, because of the learning mechanism, we use qit to show the proportion of people who choose a strategy PSi with time, and ∑i=1nqit=1.
qit is a time constant, for the first derivation, and the dynamic replication rate can be expressed as the following formula:(14)UPq=dqitdt=qUPSi−UP¯.
Similarly, the expected payoffUASi and average payoff UA¯ of different protect strategy are calculated:(15)UAS1=q1a11+q2a21+⋯+qnan1,UAS2=q1a12+q2a22+⋯+qnan2,⋮UASm=q1a1m+q2a2m+⋯+qnanm,(16)UA¯=p1UAS1+p2UAS2+⋯+pnUASm.
In the strategy setAS1,AS2,…,ASm, pit is the proportion of people who choose the attack strategy ASi, and ∑i=1mpit=1.
For a strategyASi, pit is a time constant, according to the expected payoff UASi and average payoff UA¯, and we can get the dynamic replication equation of attack strategy:(17)UAp=dpitdt=pUASi−UA¯.
By combining formulas (14) and (17), we can get the following formula:(18)UPq=dqitdt=qUPSi−UP¯,UAp=dpitdt=pUASi−UA¯.
WhenUPq=0UAp=0, we can get the equilibrium point of evolutionary game and thus realize the analysis and prediction of strategy selection.
According to the basic theory of game theory, pure strategy can be regarded as the mixed strategy with the choice probability 1 and the choice probability 0 of other strategies. Because the attack and protect game of cloud network system is a limited game, there must be a mixed strategypi∗,qj∗ to form the Nash equilibrium, and pi∗,qj∗ meets the following conditions:(19)∀pi,∑i=1m∑j=1npi∗qj∗aij≥∑i=1m∑j=1npiqj∗aij,∀qj,∑i=1m∑j=1npi∗qj∗bij≥∑i=1m∑j=1npi∗qjbij,∑i=1mpi=1,pi≥0,∑j=1nqi=1,qj≥0.


### 3.3. Protect Strategy Selection Algorithm

Both adversary and protector choose different strategy with different probabilities. We design an optimal privacy strategy selection algorithm (Algorithm1).Algorithm 1: Optimal privacy strategy selection.
Input: cloud computing attack-protect game tree.Output: optimal privacy protect strategy.(1)
InitializationAPEGM=G,S,T,U(2)
Constructing protector’s space setUP=PSi,i≥1(3)
Constructing an optional strategy space set for protectorsPS=PS1,PS2,…,PSn(4)
According to the strategy selection of the adversary, a reasonable protect strategyPSi is selected by probability qi1≤i≤m, and ∑i=1mqit=1(5)
For attack –protect strategyASi,PSj, we get the payoff value bij of protect strategy.(6)
CalculatingUPSi=p1b1i+p2b2i+⋯+pnbni of the protect strategy.(7)
Calculating the average payoffs of the protectorUP¯=∑i=1nqiUPSi.(8)
Establishing protector’s dynamic replication equationUPq=dqit/dt=qUPSi−UP¯.(9)
Computing equilibrium solution ofUPq=0.(10)
Output privacy protect strategy.From the analysis of Section3, the time complexity of the algorithm is Om+n2, and the storage space complexity is Omn. According to this algorithm, not only the payoff value of each strategy can be obtained, but also the state change rate of strategy selection can be obtained.


## 3.1. Privacy Metric Space

To make the expression of privacy information, this paper proposes a measurement model and proposes the distance between any elements in the metric space.Definition 1.
Suppose thatℜ is a nonempty set metric space; for two elements x, y in the ℜ, Φx,y represents the distance between two elements, which has two characters:(1)
Φx,y≥0, Φx,y=0, and x=y(2)
Ifz∈ℜ, Φx,y<Φx,z+Φy,z
Φx,y is the distance between two points x, y. ℜ,Φ is called as metric space according to the distance.
According to the definition of metric space, the following properties can be obtained:(1)Φx,y=Φy,x,x,y,z∈ℜ,Φx,z−Φy,z<Φx,y.
Norm is a basic concept in performance analysis, which is often used to measure the length or size of each element in the metric space.
Letnn1,n2,…,nk be a vector, and Η=hi,jm×n is a matrix.
Vector 1-norm is(2)n1=∑i=1kni.
Vector 2-norm is(3)n2=∑i=1kni21/2.
Matrix F-norm is(4)HF=∑i=1m∑j=1nhi,j21/2.
In this paper, the privacy vector is considered as an index in the measurement space, 2-norm is used to represent the size of the privacy value.
Assuming that a piece of privacy information isPIi=g1,g2,…,gi,…,gn, wi represents the privacy factor related to the user’s privacy; then the privacy value PIi of PIi can be expressed as(5)PIi=w1g12+w2g22+⋯+wngn22=∑i=1nwi×gi22,i=1,2,…,n.wi is the weight coefficient of the influence factor gi.
The correlation coefficient is an objective weight method to eliminate the influence of duplicate information on the comprehensive evaluation results. Calculate the correlation coefficient matrix; the original data containsn factors. Then the correlation coefficient matrix is(6)E=e11e12⋯e1ne21e22⋯e2n⋮⋮⋱⋮en1en2⋯enn.
After standardization, it can be simplified as(7)E=eijnn.
Calculate the sum value of1−rij in the jth column as follows:(8)∑1−ei1,∑1−ei2,…,∑1−eij.
The result of∑1−eij row vector is larger, the influence in the comprehensive evaluation system is greater, and the weight is more. Both privacy variability and conflict factor need to be considered in weight. In the paper, the objective weight method is adopted, and factor variability is represented by the standard deviation to show the difference between evaluation schemes of a factor. The larger the standard deviation is, the larger the value is.
Assume thatIi is the information quantity of ith factor; the various results of indicators can be considered by selecting the standard deviation σi. The conflict characteristics between the ith standard and other standards are measured by ∑1−eij. eij represents the correlation coefficient between the ith and the jth factors, and Ii can be expressed as follows:(9)Ii=σi∑i=1n1−eij.
The result ofIi is larger, the amount of data in the ith criterion is larger, and the importance is more. Therefore, the weight wi of the ith factor is as follows:(10)wi=Ii∑i=1′nIi.


## 3.2. Attack-Protect Evolutionary Game Model

In order to protect privacy informationPIs=PI1,PI2,…,PIn, we propose an attack-protect game model (Figure 2).Figure 2
Basic game tree of adversary-protector.Definition 2.
An attack-protect evolutionary game model can be defined as a 4-tupleAPEGM=G,S,T,U.
G=GP,GA is the participant space of evolutionary game, GP is the protector, and GA is the adversary.
S=PS,AS is game action strategy space, PS=PS1,PS2,…,PSn represents an optional strategy set for the protector, and AS=AS1,AS2,…,ASm represents an optional strategy set for the adversary. Both adversary and protector have multiple strategies.
T=pi,qi is a set of game beliefs, qi represents the probability set that the protector chooses a strategy PSi, and pi represents the probability set that the adversaries choose a strategy ASi.
U=UP,UA is a set of game payoff functions, UP represents the game payoffs of protectors, and UA expresses the game payoffs of adversaries.
In the game, both protector and adversary havePS=PS1,PS2,…,PSn and AS=AS1,AS2,…,ASm, respectively, m,n∈N, m,n≥2. When different strategies are used in the attack-protect game, both aij and bij represent the payoffs of adversary and protector when they adopt ASi and PSi. We can get the following formula:(11)a11,b11a12,b12⋯a1n,b1na21,b21a22,b22⋯a2n,b2n⋮⋮⋱⋮am1,bm1am2,bm1⋯amn,bmn.
In thePS=PS1,PS2,…,PSn, the player chooses PSi with qi, and q1+q2+⋯+qn=1. Similarly, in the AS=AS1,AS2,…,ASm, adversary selects ASi with pi, p1+p2+⋯+pm=1. Further, we can get the expected payoff UPSi and average payoff UP¯ of different protection strategies:(12)UPS1=p1b11+p2b21+⋯+pmbm1,UPS2=p1b12+p2b22+⋯+pmbm2,⋮UPSn=p1b1n+p2b2n+⋯+pmbmn,(13)UP¯=q1UPS1+q2UPS2+⋯+qnUPSn.
In strategy setPS=PS1,PS2,…,PSn, because of the learning mechanism, we use qit to show the proportion of people who choose a strategy PSi with time, and ∑i=1nqit=1.
qit is a time constant, for the first derivation, and the dynamic replication rate can be expressed as the following formula:(14)UPq=dqitdt=qUPSi−UP¯.
Similarly, the expected payoffUASi and average payoff UA¯ of different protect strategy are calculated:(15)UAS1=q1a11+q2a21+⋯+qnan1,UAS2=q1a12+q2a22+⋯+qnan2,⋮UASm=q1a1m+q2a2m+⋯+qnanm,(16)UA¯=p1UAS1+p2UAS2+⋯+pnUASm.
In the strategy setAS1,AS2,…,ASm, pit is the proportion of people who choose the attack strategy ASi, and ∑i=1mpit=1.
For a strategyASi, pit is a time constant, according to the expected payoff UASi and average payoff UA¯, and we can get the dynamic replication equation of attack strategy:(17)UAp=dpitdt=pUASi−UA¯.
By combining formulas (14) and (17), we can get the following formula:(18)UPq=dqitdt=qUPSi−UP¯,UAp=dpitdt=pUASi−UA¯.
WhenUPq=0UAp=0, we can get the equilibrium point of evolutionary game and thus realize the analysis and prediction of strategy selection.
According to the basic theory of game theory, pure strategy can be regarded as the mixed strategy with the choice probability 1 and the choice probability 0 of other strategies. Because the attack and protect game of cloud network system is a limited game, there must be a mixed strategypi∗,qj∗ to form the Nash equilibrium, and pi∗,qj∗ meets the following conditions:(19)∀pi,∑i=1m∑j=1npi∗qj∗aij≥∑i=1m∑j=1npiqj∗aij,∀qj,∑i=1m∑j=1npi∗qj∗bij≥∑i=1m∑j=1npi∗qjbij,∑i=1mpi=1,pi≥0,∑j=1nqi=1,qj≥0.


## 3.3. Protect Strategy Selection Algorithm

Both adversary and protector choose different strategy with different probabilities. We design an optimal privacy strategy selection algorithm (Algorithm1).Algorithm 1: Optimal privacy strategy selection.
Input: cloud computing attack-protect game tree.Output: optimal privacy protect strategy.(1)
InitializationAPEGM=G,S,T,U(2)
Constructing protector’s space setUP=PSi,i≥1(3)
Constructing an optional strategy space set for protectorsPS=PS1,PS2,…,PSn(4)
According to the strategy selection of the adversary, a reasonable protect strategyPSi is selected by probability qi1≤i≤m, and ∑i=1mqit=1(5)
For attack –protect strategyASi,PSj, we get the payoff value bij of protect strategy.(6)
CalculatingUPSi=p1b1i+p2b2i+⋯+pnbni of the protect strategy.(7)
Calculating the average payoffs of the protectorUP¯=∑i=1nqiUPSi.(8)
Establishing protector’s dynamic replication equationUPq=dqit/dt=qUPSi−UP¯.(9)
Computing equilibrium solution ofUPq=0.(10)
Output privacy protect strategy.From the analysis of Section3, the time complexity of the algorithm is Om+n2, and the storage space complexity is Omn. According to this algorithm, not only the payoff value of each strategy can be obtained, but also the state change rate of strategy selection can be obtained.


## 4. Improvement of the Game Model

The evolutionary game originates from the idea of biological evolution, and replication dynamic application is the most widely used in many ways. However, in the actual process of network attack and protection, not only the strategy groups of the attacker and the protector are dependent on each other, but also the protect strategy and the attack strategy are dependent on each other. Therefore, it is necessary to study the interaction among the same strategy group. In this section, we introduce the relation function to express the strategy dependence to improve the traditional replication dynamic equation and the accuracy of the replication dynamic rate.


### 4.1. Improvement of the Replication Dynamic Model

Based on Section3, in the strategy PS=PS1,PS2,…,PSn, assume that the number of strategy PSi is xit at t; the proportion of the number of protect strategy is qit, and we can get the following formula:(20)qit=xit∑i=1nxit.With the advance of the game process, the numberxit of players selecting strategy PSi changes with time, and the replication dynamic rate is proportional to the number of selection strategy PSi, which is related to the adaptability of strategy PSi. The expected payoff of protect strategy PSi is UPSi, and the average payoff of protect strategy is UP¯. We can construct the following formula:(21)xi′t=aixitUPSi,αiαi>0 is the influence factor, which indicates the impact of protect strategy PSi. The greater the value of αi, the stronger the influence of PSi on the other protect strategy. Based on formula (21), we can obtain formula (22) of PSi:(22)qi′t=xi′t∑i=1nxit−xit∑i=1nxi′t∑i=1nxit2=aiqitUPSit−UP¯t+∑j=1n1−ajaixjtUPSjt.Similarly, assume that the number ofASi is yit at time t. The proportion of the number of adversary is pit, the expected payoff is UASi, and the average payoff is UA¯. We can get the following formula to express the pit:(23)pit=yit∑i=1myit.With the advance of the attack-protect game, the number of adversary who chooses the strategyASi changes with time, and the dynamic replication rate is proportional to the number of players, so we can get the following formula:(24)yi′t=βiyitUASi.βiβi>0 is the strategy influence factor, which is determined by the strategy ASi. The larger the value of βi, the stronger the influence of ASi on other attack strategies.Furthermore, we can get the replication dynamics ofASi:(25)pi′t=yi′t∑i=1myit−yit∑i=1myi′t∑i=1myit2=βipitUASit−UA¯t+∑j=1n1−βjβiyjtUASjt.By combining formulas (22) and (25), we obtain an improved replicated dynamic differential equation:(26)qi′t=αiqitUPSit−UP¯t+∑j=1n1−αjαixjtUPSjt,pi′t==βipitUASit−UA¯t+∑j=1n1−βjβiyjtUASjt.Whenai=aj=1 and βi=βj=1, we can obtain the following dynamic replication equations:(27)qi′t=qitUPSit−UP¯t,pi′t=pitUASit−UA¯t.We define an incentive coefficientθij=ai/aj to represent incentive relationship between PSi and PSj, θij<1 indicates that protect strategy PSi can promote PSj, and θij>1 indicates that protect strategy PSi can suppress PSi.Similarly, we define an incentive coefficientΔij=βi/βj to represent incentive relationship between ASi and ASi. Through further deduction, we can get the following formula:(28)qi′t=αiqitUPSit−UP¯t+∑j=1n1−θjixjtUPSjt,pi′t==βipitUASjt−UA¯t+∑j=1n1−ΔjiyjtUASjt.Whenqi′t=0pi′t=0, we can get the equilibrium solution of attack-protect evolutionary game to realize the analysis and prediction of strategy selection.


### 4.2. Evolutionary Game Description

Assume that the protector has an optional strategy setPS1,PS2, where PS1 represents investment privacy protection and PS2 represents noninvestment privacy protection. Similarly, an adversary has an optional strategy set AS1,AS2, where AS1 represents that the adversary implements attack and AS2 represents the idea that the adversary does not implement an attack. p1 denotes the selection probability of attack strategy AS1, p2 denotes the selection probability of attack strategy AS2, and p1+p2=1; q1 denotes the selection probability of strategy AS1 and q2 denotes the selection probability of strategy AS2, and q1+q2=1.


### 4.3. Evolutionary Game Solution of Attack-Protect

Based on the attack-protect model of Section3, similarly, we can construct the following formula:(29)UPS1t=q1tb11+q2tb21,UPS2t=q1tb12+q2tb22,UP¯t=q1tUPS1t+q2tUPS2t,UAS1t=p1tb11+p2tb21,UAS2t=p1tb12+p2tb22,UA¯t=q1tUAS1t+q2tUAS2t.According toq1t+q2t=1, p1t+p2t=1, by first derivative of time t, we can get the following formula:(30)q1′t=−q2′t,p1′t=−p2′t.According to formulas (28) and (30), we can further obtain the dynamic replication equation of PS1 and AS2:(31)dq1tdt=a1q11−q1b21−θ21b22+b11−b21−θ21b21+θ21b22p1,dp1tdt=β1p11−p1a12−Δ21a22+a11−a12−Δ21a21+Δ21a22q1.Whenq1′t=0p1′t=0, we can get five solutions: q1=0p1=0, q1=0p1=1, q1=1p1=0, q1=1p1=1, and q∗=−b21+θ21b22/b11−b21−θ21b12+θ21b22p∗=−a12+Δ21a22/a11−a12−θ21a21+Δ21a22.


### 4.4. Dynamic Analysis of Attack-Protect Evolution

According to the above improved evolutionary game model, we use the Jacobian matrix method to analyze the evolutionary stability of these above five equilibrium points, and get formula (32). Both determinant det and trace tr of the Jacobian matrix can be expressed as in formulas (33) and (34):(32)J=a11−2q1b21−θ21b22+b11−b21−θ21b21+θ21b22p1a1q11−q1b11−b21−θ21a12+θ21b22β1p11−p1a11−a12−Δ21a21+Δ21a22β11−2p1a12−γ21a22+a11−a12−Δ21a21+Δ21a22q1,(33)detJ=a11−2q1b21−θ21b22+b11−b21−θ21b12+θ21b22p1β11−2p1a12−Δ21a22+a11−a12−Δ21a21+Δ21a22q1−a1q11−q1b11−b21−θ21b12+θ21b22β1p11−p1a11−a12−Δ21a21+Δ21a22,(34)trJ=a11−2q1b21−θ21b22+b11−b21−θ21b22+θ21b22p1+β11−2p1a12−Δ21a22+a11−a12−Δ21a11+Δ21a22q1.WhendetJ>0 and trJ>0, the equilibrium point is unstable; when detJ<0 and trJ is an arbitrary value, the equilibrium point is a saddle point.Condition 1.
b11−b21−θ21b21+θ21b22=0,a11−a12−Δ21a21+Δ21a22≠0; the game system has four equilibrium points: O0,0, A1,0, B1,1, C0,1. By the four points of formulas (32) and (33), we get the expressions of Table 2, and discuss several cases by determinant and trace of the Jacobian matrix:Case 1: whenb11−θ21b12<0, a11−Δ21a21<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, B1,1 is an unstable point, and (no protect, no attack) is the stable strategyCase 2: whenb11−θ21b12<0, a11−Δ21a21>0, C0,1 is a stable point, both O0,0 and B1,1 are saddle points, A1,0 is an unstable point, and (no protect, no attack) is the stable strategyCase 3: whenb11−θ21b12>0, a11−Δ21a21<0, A1,0 is a stable point, both O0,0 and B1,1 are saddle points, C0,1 is an unstable point, at this time, and (protect, no attack) is the stable strategyCase 4: whenb11−θ21b12>0, a11−Δ21a21>0, B1,1 is a stable point, both C0,1 and A1,0 are saddle points, O0,0 is an unstable point, (protect, attack) is the stable strategy, and privacy protect is the optimal selection of protectorTable 2
Determinant and trace of the equilibrium point in the game system.
EquilibriumDetTrO0,0detJ=α1β1b21−θ21b22a12−Δ21a22trJ=a1b21−θ21b22+β1a12−Δ21a22A1,0detJ=−α1β1b21−θ21b22a11−Δ21a21trJ=−a1b21−θ21b22+β1a11−Δ21a21B1,1detJ=α1β1b21−θ21b12a11−Δ21a21trJ=−a1b11−θ21b12−β1a11−Δ21a22C0,1detJ=α1β1b21−θ21b12a12−Δ21a22trJ=a1b11−θ21b12−β1a12−Δ21a22Condition 2.
Whenb11−b21−θ21b21+θ21b22=0,a11−a12−Δ21a21+Δ21a22≠0, there are four equilibrium points: O0,0, A1,0, B1,1, and C0,1; there are eight cases as follows:Case 1:b11−θ21b12<0, a11−Δ21a21<0, a12−Δ21a22<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, and B1,1 is an unstable pointCase 2:b11−θ21b12<0, a11−Δ21a21<0, a12−Δ21a22>0, A1,0 is a stable point, both O0,0 and C0,1 are saddle points, and B1,1 is an unstable pointCase 3:b11−θ21b12<0, a11−Δ21a21>0, a12−Δ21a22<0, C0,1 is a stable point, and O0,0,A1,0, and B1,1 are saddle pointsCase 4:b11−θ21b12<0, a11−Δ21a21>0, a12−Δ21a22>0, O0,0 is an unstable point, both B1,1 and A1,0 are saddle points, and C0,1 is a stable pointCase 5:b11−θ21b12>0, a11−Δ21a21<0, a12−Δ21a22<0, O0,0, C0,1, and B1,1 are saddle points, and A1,0 is an unstable pointCase 6:b11−θ21b12>0, a11−Δ21a21<0, a12−Δ21a22>0, A1,0 is a stable point, both O0,0 and B1,1 are saddle points, and C0,1 is an unstable pointCase 7:b11−θ21b12>0, a11−Δ21a21>0, a12−Δ21a22<0, B1,1 is a stable point, both O0,0 and C0,1 are saddle points, and A1,0 is an unstable pointCase 8:b11−θ21b12>0, a11−Δ21a21<0, a12−Δ21a22>0, B1,1 is a stable point, both C0,1 and A1,0 are saddle points, and O0,0 is an unstable pointCondition 3.
Whenb11−b21−θ21b21+θ21b22≠0,a11−a12−Δ21a21+Δ21a22=0, there are four equilibrium points; O0,0, A1,0, B1,1, C0,1, which can be divided into the following eight cases. Case 1:b11−θ21b12<0, a11−Δ21a21<0, b21−θ21b22<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, and B1,1 is an unstable pointCase 2:b11−θ21b12<0, a11−Δ21a21<0, b21−θ21b22>0, C0,1 is a stable point, both B1,1 and A1,0 are saddle points, and O0,0 is an unstable pointCase 3:b11−θ21b12<0, a11−Δ21a21>0, b21−θ21b22<0, O0,0 is a stable point, and C0,1, B1,1, and A1,0 are saddle pointsCase 4:b11−θ21b12<0, a11−Δ21a21>0, b21−θ21b22>0, O0,0 is a stable point, A1,0 is a saddle point, and both A1,0 and B1,1 are unstable pointsCase 5:b11−θ21b12>0, a11−Δ21a21<0, b21−θ21b22<0, A1,0 is a stable point, both O0,0 and B1,1 are saddle points, C0,1 is an unstable pointCase 6:b11−θ21b12>0, a11−Δ21a21<0, b21−θ21b22>0, O0,0 is an unstable point, both C0,1 and B1,1 are saddle points, and A1,0 is a unstable pointCase 7:b11−θ21b12>0, a11−Δ21a21>0, b21−θ21b22<0, B1,1 is a stable point, both O0,0 and A1,0 are saddle points, C0,1 is an unstable pointCase 8:b11−θ21b12>0, a11−Δ21a21>0, b21−θ21b22>0, B1,1 is a stable point, both A1,0 and C0,1 are saddle points, and O0,0 is an unstable pointCondition 4.
Whenb11−b21−θ21b12+θ21b22≠0,a11−a12−Δ21a21+Δ21a22≠0, there are five equilibrium points; O0,0, A1,0, B1,1, C0,1, q∗=−a12+Δ21a22/a11−a12−Δ21a21+Δ21a22,p∗=−b21+θ21b22/b11−b21−θ21b12+θ21b22.
Because the value ofDp∗,q∗ cannot be determined, we give several discussion of different values of q∗ and p∗:(1)
b21−θ21b22/b21−θ21b22−b11−θ21b12<0,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<0; there are four cases:Case 1:a11−Δ21a21<a12−Δ21a22<0,b11−θ21b12<b21−θ21b22<0, O0,0 is a stable point, C0,1, A1,0, and Dp∗,q∗ are saddle points, and B1,1 is an unstable pointCase 2:0<a12−Δ21a22<a11−Δ21a21,0<b21−θ21b22<b11−θ21b12, O0,0 is an unstable point, C0,1, A1,0, and Dp∗,q∗ are saddle points, and B1,1 is a stable pointCase 3:0<a12−Δ21a22<a11−Δ21a21,b11−θ21b12<b21−θ21b22<0, C0,1 is an unstable point, both P10,0 and B1,1 are saddle points, A1,0 is a stable point, and Dp∗,q∗ is a center pointCase 4:a11−Δ21a21<a12−Δ21a22<0,0<b21−θ21b22<b11−θ21b12, C0,1 is a stable point, both O0,0 and B1,1 are saddle points, A1,0 is an unstable point, and Dp∗,q∗ is a center point(2)
0<b21−θ21b22/b21−θ21b22−b11−θ21b12<1,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<0; there are four cases:Case 1:0<a12−Δ21a22<a11−Δ21a21,b11−θ21b12<0<b21−θ21b22, O0,0 is an unstable point, C0,1, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is a stable pointCase 2:a11−Δ21a21<a12−Δ21a22<0,b21−θ21b22<0<b11−θ21b12, O0,0 is a stable point, C0,1, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable pointCase 3:a11−Δ21a21<a12−Δ21a22<0,b11−θ21b12<0<b21−θ21b22, B1,1 is an unstable point, both O0,0 and A1,0 are saddle points, C0,1 is a stable point, and Dp∗,q∗ is a center pointCase 4:0<a12−Δ21a22<a11−Δ21a21,b21−θ21b22<0<b11−θ21b12, B1,1 is a stable point, both O0,0 and A1,0 are saddle points, C0,1 is an unstable point, and Dp∗,q∗ is a center point(3)
b21−θ21b22/b21−θ21b22−b11−θ21b12>1,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<0; there are four cases:Case 1:0<a12−Δ21a22<a11−Δ21a21,0<b11−θ21b12<b21−θ21b22, O0,0 is an unstable point, both C0,1 and A1,0 are saddle points, B1,1 is a stable point, and Dp∗,q∗ is a center pointCase 2:a11−Δ21a21<a12−Δ21a22<0,0<b11−θ21b12<b21−θ21b22, C0,1 is a stable point, O0,0, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable pointCase 3:0<a12−Δ21a22<a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, A1,0 is an unstable point, O0,0, B1,1, and Dp∗,q∗ are saddle points, and C0,1 is a stable pointCase 4:0>a12−Δ21a22>a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, B1,1 is an unstable point, and Dp∗,q∗ is a center point(4)
b21−θ21b22/b21−θ21b22−b11−θ21b12<0,0<a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<1; there are four cases:Case 1:a11−Δ21a21<0<a12−Δ21a22<,0<b21−θ21b22<b11−θ21b12, O0,0 is an unstable point, C0,1 is a stable point, A1,0, B1,1, and Dp∗,q∗ are saddle pointsCase 2:a12−Δ21a22<0<a11−Δ21a21,0<b21−θ21b22<b11−θ21b12, both O0,0 and C0,1 are saddle points, A1,0 is an unstable point, B1,1 is a stable point, and Dp∗,q∗ is a center pointCase 3:a12−Δ21a22<0<a11−Δ21a21,0>b21−θ21b22>b11−θ21b12, C0,1 is an unstable point, Dp∗,q∗ and both B1,1 and A1,0 are saddle points, and O0,0 is a stable pointCase 4:a11−Δ21a21<0<a12−Δ21a22,b11−θ21b12<b21−θ21b22<0, A1,0 is a stable point, both O0,0 and C0,1 are saddle points, B1,1 is an unstable point, and Dp∗,q∗ is a center point(5)
0<b21−θ21b22/b21−θ21b22−b11−θ21b12<1,0<a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<1; there are four cases:Case 1:a11−Δ21a21<0<a12−Δ21a22,b21−θ21b22<b11−θ21b12, both O0,0 and B1,1 are unstable points, both C0,1 and A1,0 are stable points, and Dp∗,q∗ is a saddle pointCase 2:a12−Δ21a22<0<a11−Δ21a21,b11−θ21b12<b21−θ21b22, both O0,0 and B1,1 are stable points, Dp∗,q∗ is a saddle point, and both C0,1 and A1,0 are unstable pointsCase 3:a12−Δ21a22<0<a11−Δ21a21,b21−θ21b22<0<b11−θ21b12, O0,0, C0,1, A1,0, and B1,1 are saddle points, and Dp∗,q∗ is a center pointCase 4:a11−Δ21a21<0<a12−Δ21a22,b11−θ21b12<0<b21−θ21b22, O0,0, C0,1, A1,0, and B1,1 are saddle points, and Dp∗,q∗ is a center point(6)
b21−θ21b22/b21−θ21b22−b11−θ21b12>1,0<a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<1; there are four cases:Case 1:a11−Δ21a21<0<a12−Δ21a22,0<b11−θ21b12<b21−θ21b22, O0,0 is an unstable point, both A1,0 and B1,1 are saddle points, C0,1 is a stable point, and Dp∗,q∗ is a center pointCase 2:a12−Δ21a22<0<a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, O0,0 is a stable point, both A1,0 and B1,1 are saddle points, C0,1 is an unstable point, and Dp∗,q∗ is a center pointCase 3:a12−Δ21a22<0<a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, B1,1 is a stable point, O0,0, C0,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable pointCase 4:a11−Δ21a21<0<a12−Δ21a22>,b21−θ21b22<b11−θ21b12<0, A1,0 is a stable point, O0,0, C0,1, and Dp∗,q∗ are saddle points, and B1,1 is an unstable point(7)
b21−θ21b22/b21−θ21b22−b11−θ21b12<0,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21>1; there are four cases:Case 1:0<a11−Δ21a21<a12−Δ21a22,0<b21−θ21b22<b11−θ21b12, O0,0 is an unstable point, both C0,1 and A1,0 are saddle points, B1,1 is a stable point, and Dp∗,q∗ is a center pointCase 2:a12−Δ21a22<a11−Δ21a21<0,b11−θ21b12<b21−θ21b22<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, B1,1 is an unstable point, and Dp∗,q∗ is a center pointCase 3:a12−Δ21a22<a11−Δ21a21<0,0<b21−θ21b22<b11−θ21b12, C0,1 is a stable point, O0,0, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable pointCase 4:a12−Δ21a22>a11−Δ21a21>0,b11−θ21b12<b21−θ21b22<0, C0,1 is a stable point, O0,0, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable point(8)
0<b21−θ21b22/b21−θ21b22−b11−θ21b12<1,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21>1; there are four cases:Case 1:0<a11−Δ21a21<a12−Δ21a22,b11−θ21b12<0<b21−θ21b22, O0,0 is an unstable point, both C0,1 and B1,1 are saddle points, A1,0 is a stable point, and Dp∗,q∗ is a center pointCase 2:a11−Δ21a21<a12−Δ21a22<0,0<b11−θ21b12<b21−θ21b22, O0,0 is a stable point, both C0,1 and B1,1 are saddle points, A1,0 is an unstable point, and Dp∗,q∗ is a center pointCase 3:a12−Δ21a22<a11−Δ21a21<0,b11−θ21b12<0<b21−θ21b22, B1,1 is an unstable point, O0,0, A1,0, and Dp∗,q∗ are saddle points, and C0,1 is a stable pointCase 4:0<a11−Δ21a21<a12−Δ21a22,b21−θ21b22<0<b11−θ21b12, B1,1 is a stable point, O0,0, A1,0, and Dp∗,q∗ are saddle points, and C0,1 is an unstable point(9)
b21−θ21b22/b21−θ21b22−b11−θ21b12>1,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21>1; there are four cases:Case 1:0<a11−Δ21a21<a12−Δ21a22,0<b11−θ21b12<b21−θ21b22, O0,0 is an unstable point, C0,1, A1,0, and Dp∗,q∗ are saddle points, and B1,1 is a stable pointCase 2:a12−Δ21a22<a11−Δ21a21<0,0>b11−θ21b12>b21−θ21b22, O0,0 is a stable point, C0,1, A1,0, and Dp∗,q∗ are saddle points, and B1,1 is an unstable pointCase 3:a12−Δ21a22<a11−Δ21a21<0,0<b21−θ21b22<b11−θ21b12, A1,0 is an unstable point, both O0,0 and B1,1 are saddle points, C0,1 is a stable point, and Dp∗,q∗ is a center pointCase 4:0<a11−Δ21a21<a12−Δ21a22,b21−θ21b22<b11−θ21b12<0, A1,0 is a stable point, both O0,0 and B1,1 are saddle points, C0,1 is an unstable point, and Dp∗,q∗ is a center point


### 4.5. Further Discussion and Analysis

According to the above analysis, when the parameters in the income matrix satisfy Conditions1–3, the system has four equilibrium points, O0,0, C0,1A1,0, and B1,1. Under these conditions, the game system has a unique evolutionary stable state, which is related to the incentive coefficients θij and Δij. When the payoff function of attack-protect satisfies Condition 4, there are five equilibrium points: O0,0, A1,0, B1,1, C0,1, and Dp∗,q∗. Next, we analyze Case 1 of situation 5 in Condition 4, and other cases are similar.Based on the above analysis, we know that, in this situation,0>a12−Δ21a22>a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, C0,1 and A1,0 are stable points, both O0,0 and B1,1 are unstable point, and Dp∗,q∗ is a saddle point. At this time, the dynamic evolution of the system is shown in Figure 3.Figure 3
The dynamic attack-protect evolutionary replication relationship ofa11−Δ21a21<0<a12−Δ21a22,b21−θ21b22<0<b11−θ21b12.In this situation,Dp∗,q∗ is an internal point in the quadrilateral of vertexes between O0,0, C0,1, A1,0, and B1,1.Based on geometry knowledge, the areaS of the quadrilateral region by O0,0, A1,0, B1,1, and Dp∗,q∗ is shown as follows:(35)S=q∗2+1−p∗2=12a12−Δ21a22a12−Δ21a22+Δ21a21−a11+θ21b12−b11b21−θ21b22+θ21b12−b11.Because of∂S/∂Δ21=a11a22−a12a21/2a12−Δ21a22+Δ21a21−a112>0, when Δ21 increases, S will increase and the equilibrium strategy is p1t=1; similarly, because of ∂S/∂θ21=b12b21−b11b22/2b21−θ21b22+θ21b12−b112<0, when θ21 increases, S will decrease and the equilibrium strategy is q1t=0. So the stable state of the system tends to the equilibrium point A1,0; on the contrary, the stable state of the system tends to the equilibrium point C0,1.This paper extends the framework of evolutionary game theory by incentive coefficient, analyzes the evolutionary stability of two players when they adopt different strategies, and reflects the influence of incentive coefficient on game decision making. Further analysis shows that the incentive coefficient can affect the optimal behavior of the players in multiple stable states and affect the evolution trend of the stable state of the dynamic system.


## 4.1. Improvement of the Replication Dynamic Model

Based on Section3, in the strategy PS=PS1,PS2,…,PSn, assume that the number of strategy PSi is xit at t; the proportion of the number of protect strategy is qit, and we can get the following formula:(20)qit=xit∑i=1nxit.With the advance of the game process, the numberxit of players selecting strategy PSi changes with time, and the replication dynamic rate is proportional to the number of selection strategy PSi, which is related to the adaptability of strategy PSi. The expected payoff of protect strategy PSi is UPSi, and the average payoff of protect strategy is UP¯. We can construct the following formula:(21)xi′t=aixitUPSi,αiαi>0 is the influence factor, which indicates the impact of protect strategy PSi. The greater the value of αi, the stronger the influence of PSi on the other protect strategy. Based on formula (21), we can obtain formula (22) of PSi:(22)qi′t=xi′t∑i=1nxit−xit∑i=1nxi′t∑i=1nxit2=aiqitUPSit−UP¯t+∑j=1n1−ajaixjtUPSjt.Similarly, assume that the number ofASi is yit at time t. The proportion of the number of adversary is pit, the expected payoff is UASi, and the average payoff is UA¯. We can get the following formula to express the pit:(23)pit=yit∑i=1myit.With the advance of the attack-protect game, the number of adversary who chooses the strategyASi changes with time, and the dynamic replication rate is proportional to the number of players, so we can get the following formula:(24)yi′t=βiyitUASi.βiβi>0 is the strategy influence factor, which is determined by the strategy ASi. The larger the value of βi, the stronger the influence of ASi on other attack strategies.Furthermore, we can get the replication dynamics ofASi:(25)pi′t=yi′t∑i=1myit−yit∑i=1myi′t∑i=1myit2=βipitUASit−UA¯t+∑j=1n1−βjβiyjtUASjt.By combining formulas (22) and (25), we obtain an improved replicated dynamic differential equation:(26)qi′t=αiqitUPSit−UP¯t+∑j=1n1−αjαixjtUPSjt,pi′t==βipitUASit−UA¯t+∑j=1n1−βjβiyjtUASjt.Whenai=aj=1 and βi=βj=1, we can obtain the following dynamic replication equations:(27)qi′t=qitUPSit−UP¯t,pi′t=pitUASit−UA¯t.We define an incentive coefficientθij=ai/aj to represent incentive relationship between PSi and PSj, θij<1 indicates that protect strategy PSi can promote PSj, and θij>1 indicates that protect strategy PSi can suppress PSi.Similarly, we define an incentive coefficientΔij=βi/βj to represent incentive relationship between ASi and ASi. Through further deduction, we can get the following formula:(28)qi′t=αiqitUPSit−UP¯t+∑j=1n1−θjixjtUPSjt,pi′t==βipitUASjt−UA¯t+∑j=1n1−ΔjiyjtUASjt.Whenqi′t=0pi′t=0, we can get the equilibrium solution of attack-protect evolutionary game to realize the analysis and prediction of strategy selection.


## 4.2. Evolutionary Game Description

Assume that the protector has an optional strategy setPS1,PS2, where PS1 represents investment privacy protection and PS2 represents noninvestment privacy protection. Similarly, an adversary has an optional strategy set AS1,AS2, where AS1 represents that the adversary implements attack and AS2 represents the idea that the adversary does not implement an attack. p1 denotes the selection probability of attack strategy AS1, p2 denotes the selection probability of attack strategy AS2, and p1+p2=1; q1 denotes the selection probability of strategy AS1 and q2 denotes the selection probability of strategy AS2, and q1+q2=1.


## 4.3. Evolutionary Game Solution of Attack-Protect

Based on the attack-protect model of Section3, similarly, we can construct the following formula:(29)UPS1t=q1tb11+q2tb21,UPS2t=q1tb12+q2tb22,UP¯t=q1tUPS1t+q2tUPS2t,UAS1t=p1tb11+p2tb21,UAS2t=p1tb12+p2tb22,UA¯t=q1tUAS1t+q2tUAS2t.According toq1t+q2t=1, p1t+p2t=1, by first derivative of time t, we can get the following formula:(30)q1′t=−q2′t,p1′t=−p2′t.According to formulas (28) and (30), we can further obtain the dynamic replication equation of PS1 and AS2:(31)dq1tdt=a1q11−q1b21−θ21b22+b11−b21−θ21b21+θ21b22p1,dp1tdt=β1p11−p1a12−Δ21a22+a11−a12−Δ21a21+Δ21a22q1.Whenq1′t=0p1′t=0, we can get five solutions: q1=0p1=0, q1=0p1=1, q1=1p1=0, q1=1p1=1, and q∗=−b21+θ21b22/b11−b21−θ21b12+θ21b22p∗=−a12+Δ21a22/a11−a12−θ21a21+Δ21a22.


## 4.4. Dynamic Analysis of Attack-Protect Evolution

According to the above improved evolutionary game model, we use the Jacobian matrix method to analyze the evolutionary stability of these above five equilibrium points, and get formula (32). Both determinant det and trace tr of the Jacobian matrix can be expressed as in formulas (33) and (34):(32)J=a11−2q1b21−θ21b22+b11−b21−θ21b21+θ21b22p1a1q11−q1b11−b21−θ21a12+θ21b22β1p11−p1a11−a12−Δ21a21+Δ21a22β11−2p1a12−γ21a22+a11−a12−Δ21a21+Δ21a22q1,(33)detJ=a11−2q1b21−θ21b22+b11−b21−θ21b12+θ21b22p1β11−2p1a12−Δ21a22+a11−a12−Δ21a21+Δ21a22q1−a1q11−q1b11−b21−θ21b12+θ21b22β1p11−p1a11−a12−Δ21a21+Δ21a22,(34)trJ=a11−2q1b21−θ21b22+b11−b21−θ21b22+θ21b22p1+β11−2p1a12−Δ21a22+a11−a12−Δ21a11+Δ21a22q1.WhendetJ>0 and trJ>0, the equilibrium point is unstable; when detJ<0 and trJ is an arbitrary value, the equilibrium point is a saddle point.Condition 1.
b11−b21−θ21b21+θ21b22=0,a11−a12−Δ21a21+Δ21a22≠0; the game system has four equilibrium points: O0,0, A1,0, B1,1, C0,1. By the four points of formulas (32) and (33), we get the expressions of Table 2, and discuss several cases by determinant and trace of the Jacobian matrix:Case 1: whenb11−θ21b12<0, a11−Δ21a21<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, B1,1 is an unstable point, and (no protect, no attack) is the stable strategyCase 2: whenb11−θ21b12<0, a11−Δ21a21>0, C0,1 is a stable point, both O0,0 and B1,1 are saddle points, A1,0 is an unstable point, and (no protect, no attack) is the stable strategyCase 3: whenb11−θ21b12>0, a11−Δ21a21<0, A1,0 is a stable point, both O0,0 and B1,1 are saddle points, C0,1 is an unstable point, at this time, and (protect, no attack) is the stable strategyCase 4: whenb11−θ21b12>0, a11−Δ21a21>0, B1,1 is a stable point, both C0,1 and A1,0 are saddle points, O0,0 is an unstable point, (protect, attack) is the stable strategy, and privacy protect is the optimal selection of protectorTable 2
Determinant and trace of the equilibrium point in the game system.
EquilibriumDetTrO0,0detJ=α1β1b21−θ21b22a12−Δ21a22trJ=a1b21−θ21b22+β1a12−Δ21a22A1,0detJ=−α1β1b21−θ21b22a11−Δ21a21trJ=−a1b21−θ21b22+β1a11−Δ21a21B1,1detJ=α1β1b21−θ21b12a11−Δ21a21trJ=−a1b11−θ21b12−β1a11−Δ21a22C0,1detJ=α1β1b21−θ21b12a12−Δ21a22trJ=a1b11−θ21b12−β1a12−Δ21a22Condition 2.
Whenb11−b21−θ21b21+θ21b22=0,a11−a12−Δ21a21+Δ21a22≠0, there are four equilibrium points: O0,0, A1,0, B1,1, and C0,1; there are eight cases as follows:Case 1:b11−θ21b12<0, a11−Δ21a21<0, a12−Δ21a22<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, and B1,1 is an unstable pointCase 2:b11−θ21b12<0, a11−Δ21a21<0, a12−Δ21a22>0, A1,0 is a stable point, both O0,0 and C0,1 are saddle points, and B1,1 is an unstable pointCase 3:b11−θ21b12<0, a11−Δ21a21>0, a12−Δ21a22<0, C0,1 is a stable point, and O0,0,A1,0, and B1,1 are saddle pointsCase 4:b11−θ21b12<0, a11−Δ21a21>0, a12−Δ21a22>0, O0,0 is an unstable point, both B1,1 and A1,0 are saddle points, and C0,1 is a stable pointCase 5:b11−θ21b12>0, a11−Δ21a21<0, a12−Δ21a22<0, O0,0, C0,1, and B1,1 are saddle points, and A1,0 is an unstable pointCase 6:b11−θ21b12>0, a11−Δ21a21<0, a12−Δ21a22>0, A1,0 is a stable point, both O0,0 and B1,1 are saddle points, and C0,1 is an unstable pointCase 7:b11−θ21b12>0, a11−Δ21a21>0, a12−Δ21a22<0, B1,1 is a stable point, both O0,0 and C0,1 are saddle points, and A1,0 is an unstable pointCase 8:b11−θ21b12>0, a11−Δ21a21<0, a12−Δ21a22>0, B1,1 is a stable point, both C0,1 and A1,0 are saddle points, and O0,0 is an unstable pointCondition 3.
Whenb11−b21−θ21b21+θ21b22≠0,a11−a12−Δ21a21+Δ21a22=0, there are four equilibrium points; O0,0, A1,0, B1,1, C0,1, which can be divided into the following eight cases. Case 1:b11−θ21b12<0, a11−Δ21a21<0, b21−θ21b22<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, and B1,1 is an unstable pointCase 2:b11−θ21b12<0, a11−Δ21a21<0, b21−θ21b22>0, C0,1 is a stable point, both B1,1 and A1,0 are saddle points, and O0,0 is an unstable pointCase 3:b11−θ21b12<0, a11−Δ21a21>0, b21−θ21b22<0, O0,0 is a stable point, and C0,1, B1,1, and A1,0 are saddle pointsCase 4:b11−θ21b12<0, a11−Δ21a21>0, b21−θ21b22>0, O0,0 is a stable point, A1,0 is a saddle point, and both A1,0 and B1,1 are unstable pointsCase 5:b11−θ21b12>0, a11−Δ21a21<0, b21−θ21b22<0, A1,0 is a stable point, both O0,0 and B1,1 are saddle points, C0,1 is an unstable pointCase 6:b11−θ21b12>0, a11−Δ21a21<0, b21−θ21b22>0, O0,0 is an unstable point, both C0,1 and B1,1 are saddle points, and A1,0 is a unstable pointCase 7:b11−θ21b12>0, a11−Δ21a21>0, b21−θ21b22<0, B1,1 is a stable point, both O0,0 and A1,0 are saddle points, C0,1 is an unstable pointCase 8:b11−θ21b12>0, a11−Δ21a21>0, b21−θ21b22>0, B1,1 is a stable point, both A1,0 and C0,1 are saddle points, and O0,0 is an unstable pointCondition 4.
Whenb11−b21−θ21b12+θ21b22≠0,a11−a12−Δ21a21+Δ21a22≠0, there are five equilibrium points; O0,0, A1,0, B1,1, C0,1, q∗=−a12+Δ21a22/a11−a12−Δ21a21+Δ21a22,p∗=−b21+θ21b22/b11−b21−θ21b12+θ21b22.
Because the value ofDp∗,q∗ cannot be determined, we give several discussion of different values of q∗ and p∗:(1)
b21−θ21b22/b21−θ21b22−b11−θ21b12<0,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<0; there are four cases:Case 1:a11−Δ21a21<a12−Δ21a22<0,b11−θ21b12<b21−θ21b22<0, O0,0 is a stable point, C0,1, A1,0, and Dp∗,q∗ are saddle points, and B1,1 is an unstable pointCase 2:0<a12−Δ21a22<a11−Δ21a21,0<b21−θ21b22<b11−θ21b12, O0,0 is an unstable point, C0,1, A1,0, and Dp∗,q∗ are saddle points, and B1,1 is a stable pointCase 3:0<a12−Δ21a22<a11−Δ21a21,b11−θ21b12<b21−θ21b22<0, C0,1 is an unstable point, both P10,0 and B1,1 are saddle points, A1,0 is a stable point, and Dp∗,q∗ is a center pointCase 4:a11−Δ21a21<a12−Δ21a22<0,0<b21−θ21b22<b11−θ21b12, C0,1 is a stable point, both O0,0 and B1,1 are saddle points, A1,0 is an unstable point, and Dp∗,q∗ is a center point(2)
0<b21−θ21b22/b21−θ21b22−b11−θ21b12<1,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<0; there are four cases:Case 1:0<a12−Δ21a22<a11−Δ21a21,b11−θ21b12<0<b21−θ21b22, O0,0 is an unstable point, C0,1, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is a stable pointCase 2:a11−Δ21a21<a12−Δ21a22<0,b21−θ21b22<0<b11−θ21b12, O0,0 is a stable point, C0,1, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable pointCase 3:a11−Δ21a21<a12−Δ21a22<0,b11−θ21b12<0<b21−θ21b22, B1,1 is an unstable point, both O0,0 and A1,0 are saddle points, C0,1 is a stable point, and Dp∗,q∗ is a center pointCase 4:0<a12−Δ21a22<a11−Δ21a21,b21−θ21b22<0<b11−θ21b12, B1,1 is a stable point, both O0,0 and A1,0 are saddle points, C0,1 is an unstable point, and Dp∗,q∗ is a center point(3)
b21−θ21b22/b21−θ21b22−b11−θ21b12>1,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<0; there are four cases:Case 1:0<a12−Δ21a22<a11−Δ21a21,0<b11−θ21b12<b21−θ21b22, O0,0 is an unstable point, both C0,1 and A1,0 are saddle points, B1,1 is a stable point, and Dp∗,q∗ is a center pointCase 2:a11−Δ21a21<a12−Δ21a22<0,0<b11−θ21b12<b21−θ21b22, C0,1 is a stable point, O0,0, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable pointCase 3:0<a12−Δ21a22<a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, A1,0 is an unstable point, O0,0, B1,1, and Dp∗,q∗ are saddle points, and C0,1 is a stable pointCase 4:0>a12−Δ21a22>a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, B1,1 is an unstable point, and Dp∗,q∗ is a center point(4)
b21−θ21b22/b21−θ21b22−b11−θ21b12<0,0<a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<1; there are four cases:Case 1:a11−Δ21a21<0<a12−Δ21a22<,0<b21−θ21b22<b11−θ21b12, O0,0 is an unstable point, C0,1 is a stable point, A1,0, B1,1, and Dp∗,q∗ are saddle pointsCase 2:a12−Δ21a22<0<a11−Δ21a21,0<b21−θ21b22<b11−θ21b12, both O0,0 and C0,1 are saddle points, A1,0 is an unstable point, B1,1 is a stable point, and Dp∗,q∗ is a center pointCase 3:a12−Δ21a22<0<a11−Δ21a21,0>b21−θ21b22>b11−θ21b12, C0,1 is an unstable point, Dp∗,q∗ and both B1,1 and A1,0 are saddle points, and O0,0 is a stable pointCase 4:a11−Δ21a21<0<a12−Δ21a22,b11−θ21b12<b21−θ21b22<0, A1,0 is a stable point, both O0,0 and C0,1 are saddle points, B1,1 is an unstable point, and Dp∗,q∗ is a center point(5)
0<b21−θ21b22/b21−θ21b22−b11−θ21b12<1,0<a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<1; there are four cases:Case 1:a11−Δ21a21<0<a12−Δ21a22,b21−θ21b22<b11−θ21b12, both O0,0 and B1,1 are unstable points, both C0,1 and A1,0 are stable points, and Dp∗,q∗ is a saddle pointCase 2:a12−Δ21a22<0<a11−Δ21a21,b11−θ21b12<b21−θ21b22, both O0,0 and B1,1 are stable points, Dp∗,q∗ is a saddle point, and both C0,1 and A1,0 are unstable pointsCase 3:a12−Δ21a22<0<a11−Δ21a21,b21−θ21b22<0<b11−θ21b12, O0,0, C0,1, A1,0, and B1,1 are saddle points, and Dp∗,q∗ is a center pointCase 4:a11−Δ21a21<0<a12−Δ21a22,b11−θ21b12<0<b21−θ21b22, O0,0, C0,1, A1,0, and B1,1 are saddle points, and Dp∗,q∗ is a center point(6)
b21−θ21b22/b21−θ21b22−b11−θ21b12>1,0<a12−Δ21a22/a12−Δ21a22−a11−Δ21a21<1; there are four cases:Case 1:a11−Δ21a21<0<a12−Δ21a22,0<b11−θ21b12<b21−θ21b22, O0,0 is an unstable point, both A1,0 and B1,1 are saddle points, C0,1 is a stable point, and Dp∗,q∗ is a center pointCase 2:a12−Δ21a22<0<a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, O0,0 is a stable point, both A1,0 and B1,1 are saddle points, C0,1 is an unstable point, and Dp∗,q∗ is a center pointCase 3:a12−Δ21a22<0<a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, B1,1 is a stable point, O0,0, C0,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable pointCase 4:a11−Δ21a21<0<a12−Δ21a22>,b21−θ21b22<b11−θ21b12<0, A1,0 is a stable point, O0,0, C0,1, and Dp∗,q∗ are saddle points, and B1,1 is an unstable point(7)
b21−θ21b22/b21−θ21b22−b11−θ21b12<0,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21>1; there are four cases:Case 1:0<a11−Δ21a21<a12−Δ21a22,0<b21−θ21b22<b11−θ21b12, O0,0 is an unstable point, both C0,1 and A1,0 are saddle points, B1,1 is a stable point, and Dp∗,q∗ is a center pointCase 2:a12−Δ21a22<a11−Δ21a21<0,b11−θ21b12<b21−θ21b22<0, O0,0 is a stable point, both C0,1 and A1,0 are saddle points, B1,1 is an unstable point, and Dp∗,q∗ is a center pointCase 3:a12−Δ21a22<a11−Δ21a21<0,0<b21−θ21b22<b11−θ21b12, C0,1 is a stable point, O0,0, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable pointCase 4:a12−Δ21a22>a11−Δ21a21>0,b11−θ21b12<b21−θ21b22<0, C0,1 is a stable point, O0,0, B1,1, and Dp∗,q∗ are saddle points, and A1,0 is an unstable point(8)
0<b21−θ21b22/b21−θ21b22−b11−θ21b12<1,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21>1; there are four cases:Case 1:0<a11−Δ21a21<a12−Δ21a22,b11−θ21b12<0<b21−θ21b22, O0,0 is an unstable point, both C0,1 and B1,1 are saddle points, A1,0 is a stable point, and Dp∗,q∗ is a center pointCase 2:a11−Δ21a21<a12−Δ21a22<0,0<b11−θ21b12<b21−θ21b22, O0,0 is a stable point, both C0,1 and B1,1 are saddle points, A1,0 is an unstable point, and Dp∗,q∗ is a center pointCase 3:a12−Δ21a22<a11−Δ21a21<0,b11−θ21b12<0<b21−θ21b22, B1,1 is an unstable point, O0,0, A1,0, and Dp∗,q∗ are saddle points, and C0,1 is a stable pointCase 4:0<a11−Δ21a21<a12−Δ21a22,b21−θ21b22<0<b11−θ21b12, B1,1 is a stable point, O0,0, A1,0, and Dp∗,q∗ are saddle points, and C0,1 is an unstable point(9)
b21−θ21b22/b21−θ21b22−b11−θ21b12>1,a12−Δ21a22/a12−Δ21a22−a11−Δ21a21>1; there are four cases:Case 1:0<a11−Δ21a21<a12−Δ21a22,0<b11−θ21b12<b21−θ21b22, O0,0 is an unstable point, C0,1, A1,0, and Dp∗,q∗ are saddle points, and B1,1 is a stable pointCase 2:a12−Δ21a22<a11−Δ21a21<0,0>b11−θ21b12>b21−θ21b22, O0,0 is a stable point, C0,1, A1,0, and Dp∗,q∗ are saddle points, and B1,1 is an unstable pointCase 3:a12−Δ21a22<a11−Δ21a21<0,0<b21−θ21b22<b11−θ21b12, A1,0 is an unstable point, both O0,0 and B1,1 are saddle points, C0,1 is a stable point, and Dp∗,q∗ is a center pointCase 4:0<a11−Δ21a21<a12−Δ21a22,b21−θ21b22<b11−θ21b12<0, A1,0 is a stable point, both O0,0 and B1,1 are saddle points, C0,1 is an unstable point, and Dp∗,q∗ is a center point


## 4.5. Further Discussion and Analysis

According to the above analysis, when the parameters in the income matrix satisfy Conditions1–3, the system has four equilibrium points, O0,0, C0,1A1,0, and B1,1. Under these conditions, the game system has a unique evolutionary stable state, which is related to the incentive coefficients θij and Δij. When the payoff function of attack-protect satisfies Condition 4, there are five equilibrium points: O0,0, A1,0, B1,1, C0,1, and Dp∗,q∗. Next, we analyze Case 1 of situation 5 in Condition 4, and other cases are similar.Based on the above analysis, we know that, in this situation,0>a12−Δ21a22>a11−Δ21a21,b21−θ21b22<b11−θ21b12<0, C0,1 and A1,0 are stable points, both O0,0 and B1,1 are unstable point, and Dp∗,q∗ is a saddle point. At this time, the dynamic evolution of the system is shown in Figure 3.Figure 3
The dynamic attack-protect evolutionary replication relationship ofa11−Δ21a21<0<a12−Δ21a22,b21−θ21b22<0<b11−θ21b12.In this situation,Dp∗,q∗ is an internal point in the quadrilateral of vertexes between O0,0, C0,1, A1,0, and B1,1.Based on geometry knowledge, the areaS of the quadrilateral region by O0,0, A1,0, B1,1, and Dp∗,q∗ is shown as follows:(35)S=q∗2+1−p∗2=12a12−Δ21a22a12−Δ21a22+Δ21a21−a11+θ21b12−b11b21−θ21b22+θ21b12−b11.Because of∂S/∂Δ21=a11a22−a12a21/2a12−Δ21a22+Δ21a21−a112>0, when Δ21 increases, S will increase and the equilibrium strategy is p1t=1; similarly, because of ∂S/∂θ21=b12b21−b11b22/2b21−θ21b22+θ21b12−b112<0, when θ21 increases, S will decrease and the equilibrium strategy is q1t=0. So the stable state of the system tends to the equilibrium point A1,0; on the contrary, the stable state of the system tends to the equilibrium point C0,1.This paper extends the framework of evolutionary game theory by incentive coefficient, analyzes the evolutionary stability of two players when they adopt different strategies, and reflects the influence of incentive coefficient on game decision making. Further analysis shows that the incentive coefficient can affect the optimal behavior of the players in multiple stable states and affect the evolution trend of the stable state of the dynamic system.


## 5. Experiment Simulation and Analysis

Based on the evolution of dynamic privacy protection game in this paper, a simple network system is deployed for simulation experiments to verify the validity of our research. The system’s topology environment is shown in Figure4, which is mainly composed of cloud computing protection device, web server, file server, data server, and client. The access control rules are that the remote host can only access the web server in the system, and the local host can access the data server.Figure 4
Experiment structural topology.We set the duration time of the game of attack and protect for 35 minutes,t∈0,35, and we can realize the optimal protect strategy selection algorithm and obtain the optimal game strategy.


### 5.1. Experiment 1

Aiming at the above conventional attack-protect model, system dynamics is used to simulate and analyze the selection of the optimal protect strategy in the evolutionary game model. Section3 shows that the evolutionary stable states analyses of this attack-protect game model are Y1=00, Y2=01, Y3=10, and Y4=11 in four kinds of situations. Next, we will simulate the different states of p and q. By observing their evolution trend, we can get the final stable state of evolution, and we can realize the prediction of attack strategy and select the optimal protect strategy:(1)
When the initial states arep=0 and q=0, the adversary chooses a strategy AS2, and the protector chooses a strategy PS2. From the evolution of the system, the choice of strategies between the adversary and the protector will be consistent. PS2 is the optimal protect strategy in Figure 5.(2)
When the initial states arep=1 and q=1, the adversary chooses a strategy AS2, and the protector chooses a strategy PS2. From the evolution of the system, it can be seen that the choice of strategies between the adversary and the protector will be consistent. PS1 is the optimal protect strategy in Figure 6.(3)
When the initial state isp=0.3 and q=0.4, the adversary chooses AS1,AS2 with the probability 0.3,0.7 and the protector chooses PS1,PS2 with the probability 0.4,0.6. After continuous evolution, the protector finally does not choose strategy PS1 and chooses PS2 to reach equilibrium state. Y1=00 is one of the evolutionary equilibrium points, and PS2 is the optimal protect strategy in Figure 7.(4)
When the initial state isp=0.6 and q=0.4, the adversary chooses AS1,AS2 with the probability 0.6,0.4, and the protector chooses PS1,PS2 with the probability 0.4,0.6. After continuous evolution, the protector finally chooses PS2 instead of PS1. Y1=01 is the equilibrium point, PS2 is the optimal protect strategy in Figure 8 at this time, (investment protection, no attack) is the evolution stable strategy of attack protect, and investment protection is the optimal selection of the protector.(5)
When the initial state isp=0.3 and q=0.7, the adversary chooses AS2, and the protector chooses a strategy PS1. After continuous evolution, the protector finally chooses PS1 to reach an equilibrium state, the value of q tends to be 1, and the value of p tends to be 0. Y3=10 is the equilibrium point, and PS1 is the optimal protect strategy in Figure 9.(6)
When the initial states arep=0.8 and q=0.7, after continuous evolution, the adversary chooses AS2, and the protector chooses strategy PS1. The values of p and q tend to be 1, Y4=11 is the evolutionary equilibrium point, and PS1 is the optimal protect strategy in Figure 10.Whenq is adjusted from 0.91 to 0.86 and p is adjusted from 0.44 to 0.38, the simulation evolution trend of the game has no obvious change in Figure 11, which indicates that the game has good robustness. Therefore, the game model has good adaptability and can be applied to complex and changeable network environment.From the above simulation results, we can see that the evolutionary system will eventually evolve to a stable state under the initial state of different strategies. By observation and comparison, we can find that the simulation results of the system are consistent with the theoretical analysis of the model, which shows that the evolutionary game model is consistent with the evolutionary law of the real system.The influence of parameter variation on system behavior is studied by parameter sensitivity test. If the parameters change slightly, the evolutionary behavior curve changes greatly, which indicates that the game model is sensitive, and it is difficult to maintain stability and robustness in the the network environment, and vice versa.Figure 5
Evolution curve ofp=0 and q=0.Figure 6
Evolution curve ofp=1 and q=1.Figure 7
Evolution curve ofp=0.3 and q=0.4.Figure 8
Evolution curve ofp=0.6 and q=0.4.Figure 9
Evolution curve ofp=0.3 and q=0.7.Figure 10
Evolution curve ofp=0.8 and q=0.7.


### 5.2. Experiment 2

During this experiment, the game strategies consist of attack strategyASi=r1,r2,…,rk and protect strategy PSi=b1,b2,…,bl. The atomic attack strategies are shown in Table 3 and the atomic protect strategy is shown in Table 4. In the experiment, we design these related attack strategies for AS1=r1,r2,r5 and AS2=r3,r4,r6 and protect strategy for PS1=b3,b4,b6 and PS2=b1,b2,b5.Table 3
Atomic attack strategy description.
NumberAttack action nameAttack strategyAS1AS2r1Weak password√r2Node attack√r3Trojan horse√r4Network monitoring√r5User name enumeration√r6Network sniffing√Table 4
Description of atomic protect strategy.
NumberProtection action nameProtect strategyPS1PS2b1System update√b2Behavioral filtering√b3Abnormal recognition√b4Delete suspicious accounts√b5Tool detection√b6Firmware update√By setting different values of incentive coefficients, we verify the influence of the dependence between different strategies on the evolution process of the game, and we highlight the superiority of the improved replicated dynamic evolutionary game model. The greater the incentive coefficient, the greater the influence among the strategies; otherwise, the smaller the influence among the strategies. In the cloud computing system, there are 200 players, according to the different values of the incentive coefficients (Table5), the initial parameters in the experiment are q1,p1=0.3,0.4 and q1,p1=0.7,0.6, and the role of different incentive coefficients in the evolution process of the game can be obtained.Table 5
The value of related game parameters.
a11=12a12=10a21=10a22=12a1=1b11=10b12=12b21=12b22=10β1=1(1)
Under the conditionsθ21=1 and Δ21=1, there is no dependency between protect strategy and attack strategy. The evolution process belongs to the classical replication dynamics. The state evolution trend of the game system is shown in Figure 12. When the initial state is q1,p1=0.3,0.4, PS1 achieves stability at the time t=7 and AS1 achieves stability at time t=8. When the initial state is q1,p1=0.7,0.6, PS1 achieves stability at time t=8 and AS1 achieves stability at time t=9.(2)
Under the conditionsθ21=0.5 and Δ21=0.5, the protect strategy PS2 promotes PS1 and the attack strategy AS2 promotes AS1. The state evolution trend of the game system is shown in Figure 13. When the initial state is q1,p1=0.3,0.4, PS1 achieves stability at time t=2 and AS1 achieves stability at time t=2. When the initial state is q1,p1=0.7,0.6, PS1 achieves stability at time t=3 and AS1 achieves stability at time t=5.(3)
Under the conditionsθ21=1.5 and Δ21=1.5, the protect strategy PS2 promotes PS1 and the attack strategy AS2 promotes AS1. The state evolution trend of the game system is shown in Figure 14. When the initial state is q1,p1=0.3,0.4, strategy PS1 achieves stability at time t=8 and strategy AS1 achieves stability at time t=10. When the initial state is q1,p1=0.7,0.6, strategy PS1 achieves stability at time t=10 and strategy AS1 achieves stability at time t=12. Compared with Figure 12, when θ21=1.5 and Δ21=1.5, different strategies in the same group have a suppression effect, which reduces the convergence speed of the protective strategy. This can lead to better performance.(4)
Under the conditionsθ21=1.5 and Δ21=0.5, the protect strategy PS2 can suppress PS1 and the attack strategy AS2 can promote AS1. The state evolution trend is shown in Figure 15. When the initial state is q1,p1=0.2,0.3, strategy PS1 achieves stability at time t=10, strategy AS1 achieves stability at time t=2. When the initial state is q1,p1=0.6,0.7, strategy PS1 achieves stability at time t=11 and strategy AS1 achieves stability at time t=4. Compared with Figure 12, when θ21=1.5, the strategy PS2 can suppress PS1 and reduce the convergence speed. When Δ21=0.5, the strategy AS2 can promote AS1 and accelerate the convergence speed.(5)
Under the conditionsθ21=0.5 and Δ21=1.5, the strategy PS2 can promote PS1 and the strategy AS2 can suppress AS1. The state evolution trend is shown in Figure 16. When the initial state is q1,p1=0.3,0.4, PS1 achieves stability at time t=2 and AS1 achieves stability at time t=10. When the initial state is q1,p1=0.7,0.6, PS1 achieves stability at time t=3 and AS1 achieves stability at time t=12. Compared with Figure 12, when Δ21=1.5, AS2 can suppress AS1 and reduce the convergence speed. When θ21=0.5, PS2 can promote PS1 and accelerate the convergence speed.Figure 11
Sensitivity test of game parameter. (a)q = 0.91, q = 0.86 and p = 0.7. (b) p = 0.44, p = 0.38, q = 0.7.
(a)(b)Figure 12
Evolution game of attack-protect underθ21=1 and Δ21=1.Figure 13
Evolution game of attack-protect underθ21=0.5 and Δ21=0.5.Figure 14
Evolution game of attack-protect underθ21=1.5 and Δ21=1.5.Figure 15
Evolution game of attack-protect underθ21=0.5 and Δ21=1.5.


### 5.3. Comparison of Several Models

According to the experimental parameters in Table5 and formulas (29) and (31), we compare this paper with [11, 14, 18] to enhance credibility, and we show the results with several experimental graphs.According to Figure17(a), with the increase of experiments, the average attack rate is stable at 0.141, 0.135, 0.125, and 0.114, respectively. Our scheme is lower than [11, 14, 18], because our scheme can moderately adjust the relationship between attack strategy and protection strategy, and has better adaptability.Figure 16
Evolution game of attack-protect underθ21=1.5 and Δ21=0.5.Figure 17
Comparison of several models.
(a)(b)(c)(d)According to Figure17(b), in terms of the average attack detection rate, alarm rate, and survived service rate, our scheme performs better than [11, 14, 18]. According to Figures 17(c) and 17(d), compared with [11, 14, 18], in our scheme, the protector can get more benefits and the attacker can get fewer benefits.Through observation and comparison, different incentive coefficients have different effects on the convergence speed of game system evolution. This shows that our research improves the evolution process of the game and can be used to guide cloud computing privacy protection.


### 5.4. Performance Comparison

To further show the characteristics and advantages of our study, the related results of comparison with other works of literature are shown in Table6.Table 6
Performance analysis and comparison.
ArticleGame typeRationalityReproduction ratePlayerEquilibriumVersatilityApplication[21]Dynamic gameComplete informationGeneral2SimplePoorCloud security[8]Dynamic gameComplete informationPoor2GeneralPoorMobile environment[17]Evolutionary gameIncomplete informationPoorNGeneralPoorStrategy selection[18]Evolutionary gameIncomplete informationPoorNGeneralGeneralStrategy selection[27]Evolutionary gameIncomplete informationPoor2GeneralGeneralCloud storage[Our]Evolutionary gameIncomplete informationGoodNGoodGoodStrategy selectionIn [8, 21], the participants are completely rational, but because of the participants’ incomplete rationality in the implementation, the feasibility of the model is reduced. Literature [17] and literature [18] are evolutionary game models, but the process of solving the model is simple. The universality of a model is mainly reflected in whether the type and strategy set are well extended to n. Paper [27] is only applicable to two game objects, and its universality is poor, which indicates that the model can only be applied to special cases. Moreover, our model has good generality by extending the optional strategy to n and can be applied to the common selection strategy.


## 5.1. Experiment 1

Aiming at the above conventional attack-protect model, system dynamics is used to simulate and analyze the selection of the optimal protect strategy in the evolutionary game model. Section3 shows that the evolutionary stable states analyses of this attack-protect game model are Y1=00, Y2=01, Y3=10, and Y4=11 in four kinds of situations. Next, we will simulate the different states of p and q. By observing their evolution trend, we can get the final stable state of evolution, and we can realize the prediction of attack strategy and select the optimal protect strategy:(1)
When the initial states arep=0 and q=0, the adversary chooses a strategy AS2, and the protector chooses a strategy PS2. From the evolution of the system, the choice of strategies between the adversary and the protector will be consistent. PS2 is the optimal protect strategy in Figure 5.(2)
When the initial states arep=1 and q=1, the adversary chooses a strategy AS2, and the protector chooses a strategy PS2. From the evolution of the system, it can be seen that the choice of strategies between the adversary and the protector will be consistent. PS1 is the optimal protect strategy in Figure 6.(3)
When the initial state isp=0.3 and q=0.4, the adversary chooses AS1,AS2 with the probability 0.3,0.7 and the protector chooses PS1,PS2 with the probability 0.4,0.6. After continuous evolution, the protector finally does not choose strategy PS1 and chooses PS2 to reach equilibrium state. Y1=00 is one of the evolutionary equilibrium points, and PS2 is the optimal protect strategy in Figure 7.(4)
When the initial state isp=0.6 and q=0.4, the adversary chooses AS1,AS2 with the probability 0.6,0.4, and the protector chooses PS1,PS2 with the probability 0.4,0.6. After continuous evolution, the protector finally chooses PS2 instead of PS1. Y1=01 is the equilibrium point, PS2 is the optimal protect strategy in Figure 8 at this time, (investment protection, no attack) is the evolution stable strategy of attack protect, and investment protection is the optimal selection of the protector.(5)
When the initial state isp=0.3 and q=0.7, the adversary chooses AS2, and the protector chooses a strategy PS1. After continuous evolution, the protector finally chooses PS1 to reach an equilibrium state, the value of q tends to be 1, and the value of p tends to be 0. Y3=10 is the equilibrium point, and PS1 is the optimal protect strategy in Figure 9.(6)
When the initial states arep=0.8 and q=0.7, after continuous evolution, the adversary chooses AS2, and the protector chooses strategy PS1. The values of p and q tend to be 1, Y4=11 is the evolutionary equilibrium point, and PS1 is the optimal protect strategy in Figure 10.Whenq is adjusted from 0.91 to 0.86 and p is adjusted from 0.44 to 0.38, the simulation evolution trend of the game has no obvious change in Figure 11, which indicates that the game has good robustness. Therefore, the game model has good adaptability and can be applied to complex and changeable network environment.From the above simulation results, we can see that the evolutionary system will eventually evolve to a stable state under the initial state of different strategies. By observation and comparison, we can find that the simulation results of the system are consistent with the theoretical analysis of the model, which shows that the evolutionary game model is consistent with the evolutionary law of the real system.The influence of parameter variation on system behavior is studied by parameter sensitivity test. If the parameters change slightly, the evolutionary behavior curve changes greatly, which indicates that the game model is sensitive, and it is difficult to maintain stability and robustness in the the network environment, and vice versa.Figure 5
Evolution curve ofp=0 and q=0.Figure 6
Evolution curve ofp=1 and q=1.Figure 7
Evolution curve ofp=0.3 and q=0.4.Figure 8
Evolution curve ofp=0.6 and q=0.4.Figure 9
Evolution curve ofp=0.3 and q=0.7.Figure 10
Evolution curve ofp=0.8 and q=0.7.


## 5.2. Experiment 2

During this experiment, the game strategies consist of attack strategyASi=r1,r2,…,rk and protect strategy PSi=b1,b2,…,bl. The atomic attack strategies are shown in Table 3 and the atomic protect strategy is shown in Table 4. In the experiment, we design these related attack strategies for AS1=r1,r2,r5 and AS2=r3,r4,r6 and protect strategy for PS1=b3,b4,b6 and PS2=b1,b2,b5.Table 3
Atomic attack strategy description.
NumberAttack action nameAttack strategyAS1AS2r1Weak password√r2Node attack√r3Trojan horse√r4Network monitoring√r5User name enumeration√r6Network sniffing√Table 4
Description of atomic protect strategy.
NumberProtection action nameProtect strategyPS1PS2b1System update√b2Behavioral filtering√b3Abnormal recognition√b4Delete suspicious accounts√b5Tool detection√b6Firmware update√By setting different values of incentive coefficients, we verify the influence of the dependence between different strategies on the evolution process of the game, and we highlight the superiority of the improved replicated dynamic evolutionary game model. The greater the incentive coefficient, the greater the influence among the strategies; otherwise, the smaller the influence among the strategies. In the cloud computing system, there are 200 players, according to the different values of the incentive coefficients (Table5), the initial parameters in the experiment are q1,p1=0.3,0.4 and q1,p1=0.7,0.6, and the role of different incentive coefficients in the evolution process of the game can be obtained.Table 5
The value of related game parameters.
a11=12a12=10a21=10a22=12a1=1b11=10b12=12b21=12b22=10β1=1(1)
Under the conditionsθ21=1 and Δ21=1, there is no dependency between protect strategy and attack strategy. The evolution process belongs to the classical replication dynamics. The state evolution trend of the game system is shown in Figure 12. When the initial state is q1,p1=0.3,0.4, PS1 achieves stability at the time t=7 and AS1 achieves stability at time t=8. When the initial state is q1,p1=0.7,0.6, PS1 achieves stability at time t=8 and AS1 achieves stability at time t=9.(2)
Under the conditionsθ21=0.5 and Δ21=0.5, the protect strategy PS2 promotes PS1 and the attack strategy AS2 promotes AS1. The state evolution trend of the game system is shown in Figure 13. When the initial state is q1,p1=0.3,0.4, PS1 achieves stability at time t=2 and AS1 achieves stability at time t=2. When the initial state is q1,p1=0.7,0.6, PS1 achieves stability at time t=3 and AS1 achieves stability at time t=5.(3)
Under the conditionsθ21=1.5 and Δ21=1.5, the protect strategy PS2 promotes PS1 and the attack strategy AS2 promotes AS1. The state evolution trend of the game system is shown in Figure 14. When the initial state is q1,p1=0.3,0.4, strategy PS1 achieves stability at time t=8 and strategy AS1 achieves stability at time t=10. When the initial state is q1,p1=0.7,0.6, strategy PS1 achieves stability at time t=10 and strategy AS1 achieves stability at time t=12. Compared with Figure 12, when θ21=1.5 and Δ21=1.5, different strategies in the same group have a suppression effect, which reduces the convergence speed of the protective strategy. This can lead to better performance.(4)
Under the conditionsθ21=1.5 and Δ21=0.5, the protect strategy PS2 can suppress PS1 and the attack strategy AS2 can promote AS1. The state evolution trend is shown in Figure 15. When the initial state is q1,p1=0.2,0.3, strategy PS1 achieves stability at time t=10, strategy AS1 achieves stability at time t=2. When the initial state is q1,p1=0.6,0.7, strategy PS1 achieves stability at time t=11 and strategy AS1 achieves stability at time t=4. Compared with Figure 12, when θ21=1.5, the strategy PS2 can suppress PS1 and reduce the convergence speed. When Δ21=0.5, the strategy AS2 can promote AS1 and accelerate the convergence speed.(5)
Under the conditionsθ21=0.5 and Δ21=1.5, the strategy PS2 can promote PS1 and the strategy AS2 can suppress AS1. The state evolution trend is shown in Figure 16. When the initial state is q1,p1=0.3,0.4, PS1 achieves stability at time t=2 and AS1 achieves stability at time t=10. When the initial state is q1,p1=0.7,0.6, PS1 achieves stability at time t=3 and AS1 achieves stability at time t=12. Compared with Figure 12, when Δ21=1.5, AS2 can suppress AS1 and reduce the convergence speed. When θ21=0.5, PS2 can promote PS1 and accelerate the convergence speed.Figure 11
Sensitivity test of game parameter. (a)q = 0.91, q = 0.86 and p = 0.7. (b) p = 0.44, p = 0.38, q = 0.7.
(a)(b)Figure 12
Evolution game of attack-protect underθ21=1 and Δ21=1.Figure 13
Evolution game of attack-protect underθ21=0.5 and Δ21=0.5.Figure 14
Evolution game of attack-protect underθ21=1.5 and Δ21=1.5.Figure 15
Evolution game of attack-protect underθ21=0.5 and Δ21=1.5.


## 5.3. Comparison of Several Models

According to the experimental parameters in Table5 and formulas (29) and (31), we compare this paper with [11, 14, 18] to enhance credibility, and we show the results with several experimental graphs.According to Figure17(a), with the increase of experiments, the average attack rate is stable at 0.141, 0.135, 0.125, and 0.114, respectively. Our scheme is lower than [11, 14, 18], because our scheme can moderately adjust the relationship between attack strategy and protection strategy, and has better adaptability.Figure 16
Evolution game of attack-protect underθ21=1.5 and Δ21=0.5.Figure 17
Comparison of several models.
(a)(b)(c)(d)According to Figure17(b), in terms of the average attack detection rate, alarm rate, and survived service rate, our scheme performs better than [11, 14, 18]. According to Figures 17(c) and 17(d), compared with [11, 14, 18], in our scheme, the protector can get more benefits and the attacker can get fewer benefits.Through observation and comparison, different incentive coefficients have different effects on the convergence speed of game system evolution. This shows that our research improves the evolution process of the game and can be used to guide cloud computing privacy protection.


## 5.4. Performance Comparison

To further show the characteristics and advantages of our study, the related results of comparison with other works of literature are shown in Table6.Table 6
Performance analysis and comparison.
ArticleGame typeRationalityReproduction ratePlayerEquilibriumVersatilityApplication[21]Dynamic gameComplete informationGeneral2SimplePoorCloud security[8]Dynamic gameComplete informationPoor2GeneralPoorMobile environment[17]Evolutionary gameIncomplete informationPoorNGeneralPoorStrategy selection[18]Evolutionary gameIncomplete informationPoorNGeneralGeneralStrategy selection[27]Evolutionary gameIncomplete informationPoor2GeneralGeneralCloud storage[Our]Evolutionary gameIncomplete informationGoodNGoodGoodStrategy selectionIn [8, 21], the participants are completely rational, but because of the participants’ incomplete rationality in the implementation, the feasibility of the model is reduced. Literature [17] and literature [18] are evolutionary game models, but the process of solving the model is simple. The universality of a model is mainly reflected in whether the type and strategy set are well extended to n. Paper [27] is only applicable to two game objects, and its universality is poor, which indicates that the model can only be applied to special cases. Moreover, our model has good generality by extending the optional strategy to n and can be applied to the common selection strategy.


## 6. Conclusion

Privacy protection has always been a hotpot in the field of cloud computing. Based on the assumption of bounded rationality, this paper proposes an attack protection game model for privacy strategy selection. Firstly, the dynamic replication equation is used to analyze the strategy process of the attack-protect and the formation mechanism of the evolution stable state. Secondly, the solution method of the evolution stable strategy is proposed, and the optimal privacy strategy selection algorithm is designed. Thirdly, aiming at the problem of strategy dependence, we construct an improved replication dynamic attack-protect evolutionary game model by incentive coefficients, and we use the Jacobian matrix to analyze the stability of the equilibrium point and get the optimal protect strategies under different conditions. Finally, the effectiveness of our research is validated by simulation experiments under several game parameters.Our research expands the evolutionary game theory and has important significance for cloud computing privacy protection [25]. Although this paper is limited to analyzing the situation of two players of two strategies, for the stable state of multiple strategies of multiple players, the same theory can be studied; only the amount of calculation is relatively complex. This will play an important role in guiding the use of evolutionary game theory to solve practical problems. Besides, there are still some shortcomings, such as the determination of attack-protect strategy set, lack of credible third-party supervision, and the quantification of incentive coefficient, which will become the future research [28, 29].



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