# Internal Resource Allocation to Information Security of Smart Cities Using Evolutionary Game Model

Authors: Jun Li; Kai Zou; Lining Xing
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

The internal resource allocation approach is proposed for the information security of smart cities using the evolutionary game model. Focusing on the information resource allocation of smart cities, the relationship among information security factors is first analyzed, then the directed connection graph and adjacency matrix are constructed to obtain its directed hierarchical structure diagram by calculating the reachable matrix, thereby developing an explanatory structure model for information security resources in smart cities. Furthermore, an evolutionary game model is established, and the replicator dynamics are used to adjust the model. Finally, three types of typical problems are investigated through simulation experiments to verify and explain the structural model.


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## Body

## 1. Introduction

As new technologies bring convenience during the process of building smart cities, there are also serious security risks, especially those related to information security. Therefore, the allocation of information resources, i.e., reasonable combination and allocation of information resources according to security requirements to achieve better security, has become an urgent issue to be solved.The main works of this study can be summarized as follows: (1) the relationship among information security factors is analyzed, then the directed connection graph and adjacency matrix are constructed to obtain its directed hierarchical structure diagram by calculating the reachable matrix, (2) the evolutionary game model is established, and the replicator dynamics are used to adjust the model, and (3) three types of typical problems are investigated through simulation experiments to verify the proposed model.


## 2. Related Works

The concept of smart cities is derived from the tenets of digital media: by utilizing various novel techniques or innovative concepts, smart cities aim to connect and integrate various systems and services in cities through the reasonable allocation of urban resources to optimize urban management and improve residents’ quality of life [1–5]. In a smart city, various novel technologies (such as the Internet of things (IoT), cloud computing, and virtual reality) are applied to different industries [6–8]. Furthermore, the city realizes the dynamically refined management and effectively improves residents’ quality of life by establishing ubiquitous broadband connectivity, integrating the application of intelligent technologies, implementing extensive resource sharing, and further attaining comprehensive and thorough enhanced ability and perception [9–11].Smart cities have been valued by countries all over the world since they came into being; they provide more convenient conditions for people’s lives, while improving the intellectual capital of cities [12–16]. As a smart city is highly dependent on some novel technical means such as cloud computing and IoT [17–24], the hidden danger of diffusion of information risk accompanies the technical application. It brings impacts from different perspectives on urban information security [25–30]. How best to allocate current urban resources and avoid the information security risk as far as possible to achieve the benefit maximization has been an inevitable practical problem faced by the healthy development of smart cities [31–36].


## 3. Influencing Factors Analysis

### 3.1. Determination of Influencing Factors

According to the requirement on the internal resource allocation to information security in smart cities, the influencing factors are categorized into ten classes (technicians (If1), management system (If2), core equipment (If3), IoT equipment (If4), network equipment (If5), application systems (If6), outside environment (If7), data (If8), preventative measures (If9), and protective measures (If10)) through comprehensive analysis.


### 3.2. Analysis of Relationship of Various Influencing Factors

According to the investigation results based on the opinions of experts in the field, the relationship between the ten classes of factors is demonstrated in Figure1.Figure 1
Relationship between influencing factors.The adjacency matrix of influencing factors can be established based on the above relationship. The matrix can express the connections between various influencing factors. According to the requirement of the interpretative structural model, the adjacency matrix of these ten classes of influencing factors (If=If1,If2,…,If10) is displayed as A=aijn×n′ in which n=10 and thus the element aij can be defined as follows:(1)aij=1,Ifishows a direct influence onIfj,0,Ifishows no direct influence onIfj,where, i,j=1,2,…,n.It can be seen from equation (1) that if the factors in the ith row directly influence those in the jth column, aij equals 1; otherwise, aij equals 0. As a result, the adjacency matrix A of various influencing factors can be derived:(2)A=If1If2If3If4If5If6If7If8If9If10If10000000011If21000000000If30001100100If40000000100If50000000100If60000000100If70011100000If80000010000If90000010100If100000010100.


### 3.3. The Directed Graph and Interpretative Structural Model

Although the above adjacency matrixA can depict the direct relationship between various influencing factors, it is difficult to reflect the indirect relationship between the factors. Therefore, it is necessary to undertake the corresponding transformation of the matrix, that is, the reachability matrix R corresponding to the adjacency matrix A is solved. The reachability matrix can clearly show the relationships of various influencing factors, whose solution process is as follows:(1)
The sumB of A and unit matrix I is calculated;(2)
The power operation is successively conducted onB until the output is constant, that is, Bk=Bk+1 and Bk≠Bk−1≠⋯≠B. In this case, the reachability matrix is R=Bk.During the power operation, it is worth noting that the power operation of the matrix should follow the operating rule of the Boolean matrix, that is,1+1=1 rather than 1+1=2; other operating rules are the same as those of the numerical operation.According to the above calculation process, it can be found thatBk=Bk+1 and Bk≠Bk−1≠⋯≠B when k=3. Thus, the reachability matrix R corresponding to the adjacency matrix is expressed as follows:(3)R=If1If2If3If4If5If6If7If8If9If10If11000010111If21100010111If30011110100If40001010100If50000110100If60000010100If70011111100If80000010100If90000010110If100000010101.According to the reachability matrix, the influencing factors are divided into a reachable setRIfi and an antecedent set AIfi which are separately expressed as RIfi=Ifj|Ifj∈If andRIfi,Ifj=1 and AIfj=Ifi|Ifi∈If andRIfi,Ifj=1 In terms of the physical meaning, the reachable set refers to a set of units that the influencing factor Ifi can reach while the antecedent set represents a set of units that can reach Ifj. After solving the reachable and antecedent sets, the bottom units can be attained, whose antecedent set only contains themselves or the strongly correlated units at the same level and the reachable set contains the other units which they can reach apart from the previous units. Therefore, all bottom units T should satisfy:(4)T=Ifi|Ifi∈If andRIfi∩AIfj=RIfi.The data in the first hierarchy can be attained according to the above description, as shown in Table1, where the bottom units that satisfy the condition are expressed as T1=If6,If8. Therefore, it can be found that application systems and data are the most direct factors influencing the information security of smart cities and they should be placed in the first hierarchy. According to the aforementioned method, the factors influencing the information security of smart cities in the second hierarchy are determined as T2=If4,If5,If9,If10 after eliminating If6 and If8. In this way, the influencing factors in the third and fourth hierarchies are separately displayed as T3=If1,If3 and T4=If2,If7. By redistributing Figure 1 according to the above hierarchy, Figure 2 can be attained: this shows the interpretative structural model for factors influencing information security in smart cities. It can be clearly found that the relationships and physical meanings of various influencing factors are quite clear on the condition that the factors are sorted according to the hierarchy.Table 1
Factors influencing the information security of smart cities in the first hierarchy.
IfiRIfiAIfiRIfi∩AIfi11, 6, 8, 9, 101, 2121, 2, 6, 8, 9, 102233, 4, 5, 6, 83, 7344, 6, 83, 4, 7455, 6, 83, 5, 7566, 81, 2, 3, 4, 5, 6, 7, 8, 9, 106, 873, 4, 5, 6, 7, 87786, 81, 2, 3, 4, 5, 6, 7, 8, 9, 106, 896, 8, 91, 2, 99106, 8, 101, 2, 1010Figure 2
The interpretative structural model for factors influencing the internal resource allocation to information security in smart cities.


### 3.4. Analysis of Hierarchical Relationship

Based on Figure2, it is found that there are four hierarchies in the interpretative structural model for factors influencing the internal resource allocation to information security in smart cities. Moreover, application systems If6 and data If8 are the most direct and fundamental factors influencing the information security of smart cities while the other factors are all taken as being indirect as they are either superficial or specific factors. Practice also proves that data and application systems are more vulnerable to illegal users, so the whole information system is not safe. Therefore, seemingly, the information security of smart cities depends on the game between illegal users and the urban information security system; it is a game of illegal users with the allocation of protective resources to data and application systems. In addition, the application systems If6 and data If8 are also influenced by IoT equipment If4, network equipment If5, preventative measures If9, protective measures If10, and core equipment If3; the preventative measures If9 and protective measures If10 are limited by the expertise of technicians If1 and the management system If2; moreover, the outside environment If7 also exerts a certain influence on the IoT equipment If4, network equipment If5, and core equipment If3. Due to the rapid development of some novel technologies including big data, cloud computing, and virtual reality, it is possible to realize cross-industry, cross-field, and cross-city data integration; however, this triggers new threats to the data; various urban application systems successively emerge, which bring convenience to people’s lives while increasing the difficulty in information security protection. Thus, more attention should be paid to data and application systems in terms of the internal resource allocation in smart cities.The 2nd to 4th hierarchies of the interpretative structural model are easily perceived: they are indirect or specific factors influencing the internal resource allocation to information security in smart cities. In terms of hardware facilities, it is evident that the core equipment, IoT equipment and network equipment can all influence the internal resource allocation to information security in smart cities. If hardware facilities are damaged, the communications and data interchange are possibly affected, even the information security and capital safety of the whole city. In terms of the defensive measures, the preventative measures, protective measures, technicians, and management systems play an important role, which are important driving forces determining internal resource allocation to information security in smart cities. With the construction of smart cities, China’s network and informatization levels are increasing; more basic work relies on information security and assurance and cities become increasingly inclusive. The better the defensive measures, the higher the level of information security. This can prevent cities from suffering huge loss, so leaders in cities are more willing to invest resources in information security. In addition, the external environment also exerts a significant influence on the safety of hardware facilities. If the workshops where facilities are placed are vulnerable to damage, for example, by earthquake, fire, or flood, the hardware facilities may fail to fulfill their intended functions. This greatly affects the data and application systems of cities, thus influencing urban information security.According to the interpretative structural model for influencing factors of the internal resource allocation to information security in smart cities, it can be found that the application systemsIf6 and data If8 are the most direct factors influencing the information security of the smart cities. They are also fundamental factors influencing the internal resource allocation to information security in smart cities. Therefore, analysis of the relational structure of factors influencing the internal resource allocation to urban information security is regarded as the premise for exploring the internal resource allocation to information security in smart cities. Through comprehensive analysis, a proper interpretative structural model is established, which provides a basis for subsequent research on the model for internal resource allocation to information security in smart cities.


## 3.1. Determination of Influencing Factors

According to the requirement on the internal resource allocation to information security in smart cities, the influencing factors are categorized into ten classes (technicians (If1), management system (If2), core equipment (If3), IoT equipment (If4), network equipment (If5), application systems (If6), outside environment (If7), data (If8), preventative measures (If9), and protective measures (If10)) through comprehensive analysis.


## 3.2. Analysis of Relationship of Various Influencing Factors

According to the investigation results based on the opinions of experts in the field, the relationship between the ten classes of factors is demonstrated in Figure1.Figure 1
Relationship between influencing factors.The adjacency matrix of influencing factors can be established based on the above relationship. The matrix can express the connections between various influencing factors. According to the requirement of the interpretative structural model, the adjacency matrix of these ten classes of influencing factors (If=If1,If2,…,If10) is displayed as A=aijn×n′ in which n=10 and thus the element aij can be defined as follows:(1)aij=1,Ifishows a direct influence onIfj,0,Ifishows no direct influence onIfj,where, i,j=1,2,…,n.It can be seen from equation (1) that if the factors in the ith row directly influence those in the jth column, aij equals 1; otherwise, aij equals 0. As a result, the adjacency matrix A of various influencing factors can be derived:(2)A=If1If2If3If4If5If6If7If8If9If10If10000000011If21000000000If30001100100If40000000100If50000000100If60000000100If70011100000If80000010000If90000010100If100000010100.


## 3.3. The Directed Graph and Interpretative Structural Model

Although the above adjacency matrixA can depict the direct relationship between various influencing factors, it is difficult to reflect the indirect relationship between the factors. Therefore, it is necessary to undertake the corresponding transformation of the matrix, that is, the reachability matrix R corresponding to the adjacency matrix A is solved. The reachability matrix can clearly show the relationships of various influencing factors, whose solution process is as follows:(1)
The sumB of A and unit matrix I is calculated;(2)
The power operation is successively conducted onB until the output is constant, that is, Bk=Bk+1 and Bk≠Bk−1≠⋯≠B. In this case, the reachability matrix is R=Bk.During the power operation, it is worth noting that the power operation of the matrix should follow the operating rule of the Boolean matrix, that is,1+1=1 rather than 1+1=2; other operating rules are the same as those of the numerical operation.According to the above calculation process, it can be found thatBk=Bk+1 and Bk≠Bk−1≠⋯≠B when k=3. Thus, the reachability matrix R corresponding to the adjacency matrix is expressed as follows:(3)R=If1If2If3If4If5If6If7If8If9If10If11000010111If21100010111If30011110100If40001010100If50000110100If60000010100If70011111100If80000010100If90000010110If100000010101.According to the reachability matrix, the influencing factors are divided into a reachable setRIfi and an antecedent set AIfi which are separately expressed as RIfi=Ifj|Ifj∈If andRIfi,Ifj=1 and AIfj=Ifi|Ifi∈If andRIfi,Ifj=1 In terms of the physical meaning, the reachable set refers to a set of units that the influencing factor Ifi can reach while the antecedent set represents a set of units that can reach Ifj. After solving the reachable and antecedent sets, the bottom units can be attained, whose antecedent set only contains themselves or the strongly correlated units at the same level and the reachable set contains the other units which they can reach apart from the previous units. Therefore, all bottom units T should satisfy:(4)T=Ifi|Ifi∈If andRIfi∩AIfj=RIfi.The data in the first hierarchy can be attained according to the above description, as shown in Table1, where the bottom units that satisfy the condition are expressed as T1=If6,If8. Therefore, it can be found that application systems and data are the most direct factors influencing the information security of smart cities and they should be placed in the first hierarchy. According to the aforementioned method, the factors influencing the information security of smart cities in the second hierarchy are determined as T2=If4,If5,If9,If10 after eliminating If6 and If8. In this way, the influencing factors in the third and fourth hierarchies are separately displayed as T3=If1,If3 and T4=If2,If7. By redistributing Figure 1 according to the above hierarchy, Figure 2 can be attained: this shows the interpretative structural model for factors influencing information security in smart cities. It can be clearly found that the relationships and physical meanings of various influencing factors are quite clear on the condition that the factors are sorted according to the hierarchy.Table 1
Factors influencing the information security of smart cities in the first hierarchy.
IfiRIfiAIfiRIfi∩AIfi11, 6, 8, 9, 101, 2121, 2, 6, 8, 9, 102233, 4, 5, 6, 83, 7344, 6, 83, 4, 7455, 6, 83, 5, 7566, 81, 2, 3, 4, 5, 6, 7, 8, 9, 106, 873, 4, 5, 6, 7, 87786, 81, 2, 3, 4, 5, 6, 7, 8, 9, 106, 896, 8, 91, 2, 99106, 8, 101, 2, 1010Figure 2
The interpretative structural model for factors influencing the internal resource allocation to information security in smart cities.


## 3.4. Analysis of Hierarchical Relationship

Based on Figure2, it is found that there are four hierarchies in the interpretative structural model for factors influencing the internal resource allocation to information security in smart cities. Moreover, application systems If6 and data If8 are the most direct and fundamental factors influencing the information security of smart cities while the other factors are all taken as being indirect as they are either superficial or specific factors. Practice also proves that data and application systems are more vulnerable to illegal users, so the whole information system is not safe. Therefore, seemingly, the information security of smart cities depends on the game between illegal users and the urban information security system; it is a game of illegal users with the allocation of protective resources to data and application systems. In addition, the application systems If6 and data If8 are also influenced by IoT equipment If4, network equipment If5, preventative measures If9, protective measures If10, and core equipment If3; the preventative measures If9 and protective measures If10 are limited by the expertise of technicians If1 and the management system If2; moreover, the outside environment If7 also exerts a certain influence on the IoT equipment If4, network equipment If5, and core equipment If3. Due to the rapid development of some novel technologies including big data, cloud computing, and virtual reality, it is possible to realize cross-industry, cross-field, and cross-city data integration; however, this triggers new threats to the data; various urban application systems successively emerge, which bring convenience to people’s lives while increasing the difficulty in information security protection. Thus, more attention should be paid to data and application systems in terms of the internal resource allocation in smart cities.The 2nd to 4th hierarchies of the interpretative structural model are easily perceived: they are indirect or specific factors influencing the internal resource allocation to information security in smart cities. In terms of hardware facilities, it is evident that the core equipment, IoT equipment and network equipment can all influence the internal resource allocation to information security in smart cities. If hardware facilities are damaged, the communications and data interchange are possibly affected, even the information security and capital safety of the whole city. In terms of the defensive measures, the preventative measures, protective measures, technicians, and management systems play an important role, which are important driving forces determining internal resource allocation to information security in smart cities. With the construction of smart cities, China’s network and informatization levels are increasing; more basic work relies on information security and assurance and cities become increasingly inclusive. The better the defensive measures, the higher the level of information security. This can prevent cities from suffering huge loss, so leaders in cities are more willing to invest resources in information security. In addition, the external environment also exerts a significant influence on the safety of hardware facilities. If the workshops where facilities are placed are vulnerable to damage, for example, by earthquake, fire, or flood, the hardware facilities may fail to fulfill their intended functions. This greatly affects the data and application systems of cities, thus influencing urban information security.According to the interpretative structural model for influencing factors of the internal resource allocation to information security in smart cities, it can be found that the application systemsIf6 and data If8 are the most direct factors influencing the information security of the smart cities. They are also fundamental factors influencing the internal resource allocation to information security in smart cities. Therefore, analysis of the relational structure of factors influencing the internal resource allocation to urban information security is regarded as the premise for exploring the internal resource allocation to information security in smart cities. Through comprehensive analysis, a proper interpretative structural model is established, which provides a basis for subsequent research on the model for internal resource allocation to information security in smart cities.


## 4. Evolutionary Game Model

### 4.1. Basic Assumptions

Assumption 1.
It is supposed that a smart city is a closed system and does not share or exchange any other resource with the external environment.Assumption 2.
The evolutional game process is dynamically interactive. The game participants can adjust their own strategies by observing and learning from the strategies of the other participants; through constant observation and adjustment, the strategy equilibrium is finally realized.Assumption 3.
All game participants are boundedly rational. Compared with the full rationality of the participants in the traditional game models, the evolutionary game model belongs to a dynamic process. The game participants only master part of others’ revenues on condition of the formulated strategy. In this case, they constantly adjust strategies to pursue revenue maximization, which conforms to actual conditions.Assumption 4.
The game process is fair. The participants constantly adjust their own strategies until reaching evolutionary equilibrium in the process of the evolutionary game. Under the evolutionary equilibrium, they select different strategies while attaining the same revenue, thus realizing a state of fairness.Assumption 5.
All resources of the city are transformed into funds and then measured; the internal resource allocation problem is converted into the fund allocation problem; the internal resources allocated to urban information security are converted into the total fundingFt.


### 4.2. Problem Modelling

A game problem can be transformed into a mathematical expressionGT=P,St,Ut.The internal resource allocation to information security in smart cities is mainly influenced by factors including technicians, the management system, core equipment, IoT equipment, network equipment, application systems, outside environment, data, and preventative and protective measures. The data and application systems show the most direct influence. In this section, only the data and application systems are taken as examples and analyzed (other influencing factors deliver a similar result). Therefore, the problem concerning the internal resource allocation to information security in a smart city can be transformed into the fund allocation problem to data and application systems.Game participants: the total amount of funds can be divided intoN equal parts, each of which FN=Ft/N makes up a game participant. r=1,2,…,N represents a set of the game participants. Moreover, it is supposed that the decision-making of each game participant is independent before adjustment. When adjusting the decision-making, it is necessary to adjust their own strategy according to the decision-makings of the other participants.Populations: A population comprises the set of game participants in the process of an evolutionary game, which is expressed as=P1,P2,…,PN.The set of strategies: The number of strategies is the same as thatM of influencing factors. Thus, the set of strategies can be expressed as St=St1,St2,…,StM. In the case of only considering data and application systems, M=2. That is, the game participants select to invest in the data St1 and application systems St2. The two influencing factors make up a set of strategies, which is shown as t=St1,St2.The population ratio: This refers to the ratio of the number of game participants who select a certain strategy in the population to the total population, i.e.Xi=ni/N, in which i∈St and ni denotes the number of game participants who select a strategy i. Obviously, ∑i=1StXi=1.The population state: This means the set of population ratios corresponding to all selected strategies. That is,X=Xi|i∈St represents all possible population states in the problem model.Loss function: Each game participant is a rational person and selects their access strategies through different losses. In the game process, the loss function of the game participants is defined as the sum of the actual loss caused by the invasion of illegal users and the invested fund. The game participants aim to minimize their loss function when selecting a strategy.According to its definition, it is found that the population ratio means the probability that game participants invest in theith influencing factor. It is supposed that the city government invests funds in the ith influencing factor and the loss reduced by unit funding is Ei; the probability that the ith influencing factor is affected by illegal users is set to pi and the probability that cities are invaded by illegal users when no funds are invested in the ith influencing factor of information security is vi; moreover, the loss caused by the intrusion of illegal users is expressed as Li. According to the model proposed by Gordon [33] and the current assumptions, the model is improved. On this basis, the probability that data are directly invaded by illegal users is psi=piviXiNFNEi+1=piviXiFtEi+1. Therefore, smart cities aim to minimize their expected cost after being invaded. Furthermore, the loss function C of smart cities can be deduced as follows:(5)C=∑i=1MpsiLi+Ft=∑i=1MpivixiFtEi+1Li+Ft,where the constraint is such that ∑i=1MXi=1.By introducing equation (5) and the constraint into the Lagrange multiplier, it can be attained that:(6)L=∑i=1MpivixiFtEi+1Li+Ft+λ∑i=1Mxi−1.Partial differentiation of equation (6) gives:(7)∂L∂xi=FtEiLipivixiFtEi+1+λ,∂L∂λ=∑i=1Mxi−1.By calculating the second-order partial derivative of equation (6), it is found that:(8)∂2L∂xi2=Ft2Ei2LipivixiFtEi+1.It can be seen from equation (8) that ∂2L/∂xi2≥0 is always valid. Thus, the minimum of the loss function C is found at ∂L/∂xi=0 and ∂L/∂λ=0. As a result, it is found that the loss of smart cities is the minimum when the Nash equilibrium solution of the problem is calculated as y∗=x1∗,x1∗,…,xM∗, in which y∗ satisfies equation (7).The set of strategies in the smart cities only considering the influences of data and application systems contains two strategies. That is, the game participants select to invest in the dataSt1 and application systems St2. The two strategies make up a set (St=St1,St2) of strategies. It is supposed that the probabilities that the game participants invest in data and application software are x1 and x2, respectively, and thus x2=1−x1. It is supposed that the losses reduced per unit funding invested in data and application software are expressed as E1 and E2; the probabilities that cities are invaded by illegal users when no funds are invested in the data and application software influencing information security are separately expressed as v1 and v2; the losses caused by invasion to data and application software are L1 and L2. According to the previous description, the probabilities that data and application software are directly invaded by illegal users are ps1=p1v1x1FtE1+1 and ps2=p2v21−x1FtE2+1, respectively. Therefore, smart cities aim to minimize their expected cost after being invaded. Furthermore, the loss function C of smart cities can be deduced as follows.It can be seen that∂2C/∂x12≥0 is always established and therefore the minimum of the loss function C appears at ∂C/∂x1=0. Hence, the Nash equilibrium solution of the problem can be attained as y∗=x1∗,x2∗, in which x1 and x2 satisfy equation:(9)x1∗=FtE2lnv2+lnE2L2p2v2−lnE1L1p1v1FtE1lnv1+E2lnv2,x2∗=1−FtE2lnv2+lnE2L2p2v2−lnE1L1p1v1FtE1lnv1+E2lnv2,0≤x1∗≤1,0≤x2∗≤1.On condition that the funds invested in data and application software arex1∗Ft and x2∗Ft, the loss of smart cities is minimized, thus resulting in optimal benefit.


## 4.1. Basic Assumptions

Assumption 1.
It is supposed that a smart city is a closed system and does not share or exchange any other resource with the external environment.Assumption 2.
The evolutional game process is dynamically interactive. The game participants can adjust their own strategies by observing and learning from the strategies of the other participants; through constant observation and adjustment, the strategy equilibrium is finally realized.Assumption 3.
All game participants are boundedly rational. Compared with the full rationality of the participants in the traditional game models, the evolutionary game model belongs to a dynamic process. The game participants only master part of others’ revenues on condition of the formulated strategy. In this case, they constantly adjust strategies to pursue revenue maximization, which conforms to actual conditions.Assumption 4.
The game process is fair. The participants constantly adjust their own strategies until reaching evolutionary equilibrium in the process of the evolutionary game. Under the evolutionary equilibrium, they select different strategies while attaining the same revenue, thus realizing a state of fairness.Assumption 5.
All resources of the city are transformed into funds and then measured; the internal resource allocation problem is converted into the fund allocation problem; the internal resources allocated to urban information security are converted into the total fundingFt.


## 4.2. Problem Modelling

A game problem can be transformed into a mathematical expressionGT=P,St,Ut.The internal resource allocation to information security in smart cities is mainly influenced by factors including technicians, the management system, core equipment, IoT equipment, network equipment, application systems, outside environment, data, and preventative and protective measures. The data and application systems show the most direct influence. In this section, only the data and application systems are taken as examples and analyzed (other influencing factors deliver a similar result). Therefore, the problem concerning the internal resource allocation to information security in a smart city can be transformed into the fund allocation problem to data and application systems.Game participants: the total amount of funds can be divided intoN equal parts, each of which FN=Ft/N makes up a game participant. r=1,2,…,N represents a set of the game participants. Moreover, it is supposed that the decision-making of each game participant is independent before adjustment. When adjusting the decision-making, it is necessary to adjust their own strategy according to the decision-makings of the other participants.Populations: A population comprises the set of game participants in the process of an evolutionary game, which is expressed as=P1,P2,…,PN.The set of strategies: The number of strategies is the same as thatM of influencing factors. Thus, the set of strategies can be expressed as St=St1,St2,…,StM. In the case of only considering data and application systems, M=2. That is, the game participants select to invest in the data St1 and application systems St2. The two influencing factors make up a set of strategies, which is shown as t=St1,St2.The population ratio: This refers to the ratio of the number of game participants who select a certain strategy in the population to the total population, i.e.Xi=ni/N, in which i∈St and ni denotes the number of game participants who select a strategy i. Obviously, ∑i=1StXi=1.The population state: This means the set of population ratios corresponding to all selected strategies. That is,X=Xi|i∈St represents all possible population states in the problem model.Loss function: Each game participant is a rational person and selects their access strategies through different losses. In the game process, the loss function of the game participants is defined as the sum of the actual loss caused by the invasion of illegal users and the invested fund. The game participants aim to minimize their loss function when selecting a strategy.According to its definition, it is found that the population ratio means the probability that game participants invest in theith influencing factor. It is supposed that the city government invests funds in the ith influencing factor and the loss reduced by unit funding is Ei; the probability that the ith influencing factor is affected by illegal users is set to pi and the probability that cities are invaded by illegal users when no funds are invested in the ith influencing factor of information security is vi; moreover, the loss caused by the intrusion of illegal users is expressed as Li. According to the model proposed by Gordon [33] and the current assumptions, the model is improved. On this basis, the probability that data are directly invaded by illegal users is psi=piviXiNFNEi+1=piviXiFtEi+1. Therefore, smart cities aim to minimize their expected cost after being invaded. Furthermore, the loss function C of smart cities can be deduced as follows:(5)C=∑i=1MpsiLi+Ft=∑i=1MpivixiFtEi+1Li+Ft,where the constraint is such that ∑i=1MXi=1.By introducing equation (5) and the constraint into the Lagrange multiplier, it can be attained that:(6)L=∑i=1MpivixiFtEi+1Li+Ft+λ∑i=1Mxi−1.Partial differentiation of equation (6) gives:(7)∂L∂xi=FtEiLipivixiFtEi+1+λ,∂L∂λ=∑i=1Mxi−1.By calculating the second-order partial derivative of equation (6), it is found that:(8)∂2L∂xi2=Ft2Ei2LipivixiFtEi+1.It can be seen from equation (8) that ∂2L/∂xi2≥0 is always valid. Thus, the minimum of the loss function C is found at ∂L/∂xi=0 and ∂L/∂λ=0. As a result, it is found that the loss of smart cities is the minimum when the Nash equilibrium solution of the problem is calculated as y∗=x1∗,x1∗,…,xM∗, in which y∗ satisfies equation (7).The set of strategies in the smart cities only considering the influences of data and application systems contains two strategies. That is, the game participants select to invest in the dataSt1 and application systems St2. The two strategies make up a set (St=St1,St2) of strategies. It is supposed that the probabilities that the game participants invest in data and application software are x1 and x2, respectively, and thus x2=1−x1. It is supposed that the losses reduced per unit funding invested in data and application software are expressed as E1 and E2; the probabilities that cities are invaded by illegal users when no funds are invested in the data and application software influencing information security are separately expressed as v1 and v2; the losses caused by invasion to data and application software are L1 and L2. According to the previous description, the probabilities that data and application software are directly invaded by illegal users are ps1=p1v1x1FtE1+1 and ps2=p2v21−x1FtE2+1, respectively. Therefore, smart cities aim to minimize their expected cost after being invaded. Furthermore, the loss function C of smart cities can be deduced as follows.It can be seen that∂2C/∂x12≥0 is always established and therefore the minimum of the loss function C appears at ∂C/∂x1=0. Hence, the Nash equilibrium solution of the problem can be attained as y∗=x1∗,x2∗, in which x1 and x2 satisfy equation:(9)x1∗=FtE2lnv2+lnE2L2p2v2−lnE1L1p1v1FtE1lnv1+E2lnv2,x2∗=1−FtE2lnv2+lnE2L2p2v2−lnE1L1p1v1FtE1lnv1+E2lnv2,0≤x1∗≤1,0≤x2∗≤1.On condition that the funds invested in data and application software arex1∗Ft and x2∗Ft, the loss of smart cities is minimized, thus resulting in optimal benefit.


## 5. Experimental Results and Analysis

Based on the simulation test, it is feasible to analyze the internal resource allocation of a smart city and further obtain the final resource allocation. Through numerical analysis, the correlation analysis is separately conducted on various influencing factors, thus acquiring the optimal resource allocation.Diverse conditions will occur during the practical construction of a smart city and it is impossible to list and analyze all conditions. Thus, only three typical conditions are analyzed: (1) only considering different probabilities that a city is invaded by illegal users when no funds are invested in the information security, that is, only considering the influences ofv1 and v2; (2) only considering different losses caused by invasion of illegal users in the city, that is, only considering the influences of L1 and L2; (3) only considering different losses reduced by the unit fund of the city after investing funds in influencing factors, that is, only considering the influences of E1 and E2.


### 5.1. Influences of Factorsv1 and v2

It is supposed that the other parameters are kept the same and only the results induced by different probabilities that a city is affected by illegal users when no funds are invested in information security are analyzed. That is, the different results under different values ofv1 and v2 are considered.The influence of investing funds in the data is slightly better than investing in the application software in actual conditions. Considering this, it is supposed that the losses reduced by separately investing 0.1 billion CNY in data and application software in smart cities areE1=8 million CNY and E2=6 million CNY. It is supposed that the probabilities of being affected by illegal users are both 0.5. The losses caused by such illegal use of the data and application software are L1=1 billion CNY and L2=1 billion CNY. The total invested funds Ft equal 10 billion CNY.According to equation (9), the Nash equilibrium solutions and expected losses under different values ofv1 and v2 can be calculated, as shown in Table 2.Table 2
The influences ofv1 and v2 on the expected loss.
v1,v2x1,x2Expected loss(0.3, 0.3)(0.4456, 0.5544)100.0479(0.3, 0.5)(0.2854, 0.7146)100.2240(0.3, 0.7)(0.1343, 0.8657)100.9604(0.5, 0.3)(0.6283, 0.3717)100.1790(0.5, 0.5)(0.4582, 0.5418)100.4596(0.5, 0.7)(0.2721, 0.7279)101.2900(0.7, 0.3)(0.8295, 0.1705)100.7659(0.7, 0.5)(0.6821, 0.3179)101.1662(0.7, 0.7)(0.4862, 0.5138)102.0397The values ofv1 and v2 are within 0.1,0.9 and therefore the expected losses of smart cities are as shown in Figure 3.Figure 3
Changes in the expected loss withv1 and v2. In this figure, x axis denotes the probabilities that cities are invaded by illegal users when no funds are invested in the data influencing information security (v1), y axis denotes the probabilities that cities are invaded by illegal users when no funds are invested in the application software influencing information security (v2), and z axis denotes expected loss of smart cities.According to Table2 and Figure 3, the following conclusions can be drawn:(1)
When fixing the probability of invasion of illegal users to one influencing factor, the expected loss of smart cities shows a positive correlation with the probability of invasion to the other factor. That is, the higher the probability of invasion by illegal users when no funds are invested in information security is, the larger the expected loss to these smart cities. It implies that the influencing factor shows a great importance in urban resource allocation. Therefore, it is necessary to increase the resources allocated to the influencing factors with a high probability of invasion; on the contrary, the lower the probability of being invaded by illegal users when no funds are invested in information security, the lower the expected loss to cities. This indicates that the influencing factor is less important in urban resource allocation and investment therein can be slightly reduced.(2)
When influencing factors have similar probabilities of being invaded, the expected loss of cities is low only when the levels of the resource allocation to the factors are also similar.


### 5.2. Influences of FactorsL1 and L2

As shown in equation (9), the influences of L1 and L2 are similar to those of P1 and P2. Therefore, only one factor is analyzed. It is supposed that the other parameters remain unchanged. The results of different losses caused by the invasion of illegal users in smart cities are only investigated. That is, the different results under different values of L1 and L2 are analyzed.The influence of investing funds in the data is better than investing in the application software in actual conditions. Considering this, it is supposed that the losses reduced by separately investing 0.1 billion CNY in data and application software in smart cities areE1=8 million CNY and E2=6 million CNY. It is supposed that the probabilities of being invaded by illegal users are both 0.5. The probabilities that cities are invaded by illegal users when no funds are invested in data and application software influencing the information security are separately shown as v1=0.3 and v2=0.5. The total invested funding Ft is 10 billion CNY.According to equation (10), the Nash equilibrium solutions and expected losses under different values ofL1 and L2 can be solved, as shown in Table 3.Table 3
The influences ofL1 and L2 on the expected loss.
L1,L2x1,x2Expected loss(30, 30)(0.2854, 0.7146)100.6720(30, 50)(0.2484, 0.7516)100.9601(30, 70)(0.2240, 0.7760)101.2145(50, 30)(0.3224, 0.6776)100.7840(50, 50)(0.2854, 0.7146)101.1200(50, 70)(0.2610, 0.7390)101.4168(70, 30)(0.3468, 0.6532)100.8677(70, 50)(0.3098, 0.6902)101.2397(70, 70)(0.2854, 0.7146)101.5681By analyzing Table3 and Figure 4, the following conclusions can be drawn: (1)
When fixing the loss caused by invasion to one influencing factor, the expected loss of cities is positively correlated with that triggered by invasion to the other factor. That is, the greater the loss caused by the invasion of illegal users in cities, the greater the expected loss, therefore, it is essential to increase the resources allocated to the influencing factor which is invaded and results in the largest loss; on the contrary, the lower the loss caused by invasion of illegal users in cities, the lower the expected loss. This indicates that the influencing factor is less important to urban resource allocation and investment therein can be slightly reduced.(2)
In the case that the losses caused by invasion are similar across influencing factors, the resource allocation to each factor remains the same. That is, the influencing factors do not affect the resource allocation to information security in cities provided the losses caused by invasion are equivalent across each factor.Figure 4
Changes of the expected loss withL1 and L2. In this figure, x axis denotes the losses caused by invasion to the data (L1), y axis denotes the losses caused by invasion to the application software (L2), and z axis denotes expected loss of smart cities.


### 5.3. Influences of FactorsE1 and E2

It is supposed that the other parameters remain unchanged and only the results of different losses reduced by the city government investing 0.1 billion CNY in influencing factors are studied. That is, different results under different values ofE1 and E2 are explored.It is supposed that the probabilities of being invaded by illegal users are both 0.5. The probabilities that a smart city is invaded by illegal users when no funds are invested in data and application software influencing the information security are separately shown asv1=0.3 and v2=0.5. The losses caused by the invasion of the data and application software are L1=1 billion CNY and L2=1 billion CNY. The total funding invested is Ft=10 a billion CNY.According to equation (10), the Nash equilibrium solutions and expected losses under different values ofE1 and E2 are calculated, as shown in Table 4.Table 4
The influences ofE1 and E2 on the expected loss.
E1,E2x1,x2Expected loss(0.03, 0.03)(0.2756, 0.7244)101.1086(0.03, 0.05)(0.3453, 0.6547)100.6895(0.03, 0.07)(0.4128, 0.5872)100.4825(0.05, 0.03)(0.2567, 0.7433)100.8528(0.05, 0.05)(0.3115, 0.6885)100.4599(0.05, 0.07)(0.3684, 0.6316)100.2800(0.07, 0.03)(0.2299, 0.7701)100.7201(0.07, 0.05)(0.2767, 0.7233)100.3495(0.07, 0.07)(0.3269, 0.6731)100.1908By analyzing Table4 and Figure 5, the following conclusions can be drawn:(1)
In the case of fixing the loss reduced by investing in one influencing factor, the expected loss of cities is negatively correlated with the loss reduced by investing in the other factor. That is, the greater the loss reduced by unit fund invested in influencing factors, the lower the expected loss from the cities. It implies that the influencing factor is less important to urban resource allocation and investment therein can be decreased slightly; on the contrary, the lower the loss reduced by unit fund invested in an influencing factor is, the larger the expected loss. This indicates that the influencing factor is of great importance to urban resource allocation and it is essential to increase the resource allocated thereto, provided the loss reduced per unit funding is low.(2)
On the condition of fixing the probabilities that the data and application software are invaded by illegal users and keepingv1<v2, the investment in the data is always low while in the application software is large in terms of the overall expectation of the resource allocation regardless of changes of the loss reduced by unit fund invested by cities in influencing factors.Figure 5
Changes in the expected loss withE1 and E2. In this figure, x axis denotes the losses reduced per unit funding invested in data (E1), y axis denotes the losses reduced per unit funding invested in application software (E2), and z axis denotes the expected loss of smart cities.


## 5.1. Influences of Factorsv1 and v2

It is supposed that the other parameters are kept the same and only the results induced by different probabilities that a city is affected by illegal users when no funds are invested in information security are analyzed. That is, the different results under different values ofv1 and v2 are considered.The influence of investing funds in the data is slightly better than investing in the application software in actual conditions. Considering this, it is supposed that the losses reduced by separately investing 0.1 billion CNY in data and application software in smart cities areE1=8 million CNY and E2=6 million CNY. It is supposed that the probabilities of being affected by illegal users are both 0.5. The losses caused by such illegal use of the data and application software are L1=1 billion CNY and L2=1 billion CNY. The total invested funds Ft equal 10 billion CNY.According to equation (9), the Nash equilibrium solutions and expected losses under different values ofv1 and v2 can be calculated, as shown in Table 2.Table 2
The influences ofv1 and v2 on the expected loss.
v1,v2x1,x2Expected loss(0.3, 0.3)(0.4456, 0.5544)100.0479(0.3, 0.5)(0.2854, 0.7146)100.2240(0.3, 0.7)(0.1343, 0.8657)100.9604(0.5, 0.3)(0.6283, 0.3717)100.1790(0.5, 0.5)(0.4582, 0.5418)100.4596(0.5, 0.7)(0.2721, 0.7279)101.2900(0.7, 0.3)(0.8295, 0.1705)100.7659(0.7, 0.5)(0.6821, 0.3179)101.1662(0.7, 0.7)(0.4862, 0.5138)102.0397The values ofv1 and v2 are within 0.1,0.9 and therefore the expected losses of smart cities are as shown in Figure 3.Figure 3
Changes in the expected loss withv1 and v2. In this figure, x axis denotes the probabilities that cities are invaded by illegal users when no funds are invested in the data influencing information security (v1), y axis denotes the probabilities that cities are invaded by illegal users when no funds are invested in the application software influencing information security (v2), and z axis denotes expected loss of smart cities.According to Table2 and Figure 3, the following conclusions can be drawn:(1)
When fixing the probability of invasion of illegal users to one influencing factor, the expected loss of smart cities shows a positive correlation with the probability of invasion to the other factor. That is, the higher the probability of invasion by illegal users when no funds are invested in information security is, the larger the expected loss to these smart cities. It implies that the influencing factor shows a great importance in urban resource allocation. Therefore, it is necessary to increase the resources allocated to the influencing factors with a high probability of invasion; on the contrary, the lower the probability of being invaded by illegal users when no funds are invested in information security, the lower the expected loss to cities. This indicates that the influencing factor is less important in urban resource allocation and investment therein can be slightly reduced.(2)
When influencing factors have similar probabilities of being invaded, the expected loss of cities is low only when the levels of the resource allocation to the factors are also similar.


## 5.2. Influences of FactorsL1 and L2

As shown in equation (9), the influences of L1 and L2 are similar to those of P1 and P2. Therefore, only one factor is analyzed. It is supposed that the other parameters remain unchanged. The results of different losses caused by the invasion of illegal users in smart cities are only investigated. That is, the different results under different values of L1 and L2 are analyzed.The influence of investing funds in the data is better than investing in the application software in actual conditions. Considering this, it is supposed that the losses reduced by separately investing 0.1 billion CNY in data and application software in smart cities areE1=8 million CNY and E2=6 million CNY. It is supposed that the probabilities of being invaded by illegal users are both 0.5. The probabilities that cities are invaded by illegal users when no funds are invested in data and application software influencing the information security are separately shown as v1=0.3 and v2=0.5. The total invested funding Ft is 10 billion CNY.According to equation (10), the Nash equilibrium solutions and expected losses under different values ofL1 and L2 can be solved, as shown in Table 3.Table 3
The influences ofL1 and L2 on the expected loss.
L1,L2x1,x2Expected loss(30, 30)(0.2854, 0.7146)100.6720(30, 50)(0.2484, 0.7516)100.9601(30, 70)(0.2240, 0.7760)101.2145(50, 30)(0.3224, 0.6776)100.7840(50, 50)(0.2854, 0.7146)101.1200(50, 70)(0.2610, 0.7390)101.4168(70, 30)(0.3468, 0.6532)100.8677(70, 50)(0.3098, 0.6902)101.2397(70, 70)(0.2854, 0.7146)101.5681By analyzing Table3 and Figure 4, the following conclusions can be drawn: (1)
When fixing the loss caused by invasion to one influencing factor, the expected loss of cities is positively correlated with that triggered by invasion to the other factor. That is, the greater the loss caused by the invasion of illegal users in cities, the greater the expected loss, therefore, it is essential to increase the resources allocated to the influencing factor which is invaded and results in the largest loss; on the contrary, the lower the loss caused by invasion of illegal users in cities, the lower the expected loss. This indicates that the influencing factor is less important to urban resource allocation and investment therein can be slightly reduced.(2)
In the case that the losses caused by invasion are similar across influencing factors, the resource allocation to each factor remains the same. That is, the influencing factors do not affect the resource allocation to information security in cities provided the losses caused by invasion are equivalent across each factor.Figure 4
Changes of the expected loss withL1 and L2. In this figure, x axis denotes the losses caused by invasion to the data (L1), y axis denotes the losses caused by invasion to the application software (L2), and z axis denotes expected loss of smart cities.


## 5.3. Influences of FactorsE1 and E2

It is supposed that the other parameters remain unchanged and only the results of different losses reduced by the city government investing 0.1 billion CNY in influencing factors are studied. That is, different results under different values ofE1 and E2 are explored.It is supposed that the probabilities of being invaded by illegal users are both 0.5. The probabilities that a smart city is invaded by illegal users when no funds are invested in data and application software influencing the information security are separately shown asv1=0.3 and v2=0.5. The losses caused by the invasion of the data and application software are L1=1 billion CNY and L2=1 billion CNY. The total funding invested is Ft=10 a billion CNY.According to equation (10), the Nash equilibrium solutions and expected losses under different values ofE1 and E2 are calculated, as shown in Table 4.Table 4
The influences ofE1 and E2 on the expected loss.
E1,E2x1,x2Expected loss(0.03, 0.03)(0.2756, 0.7244)101.1086(0.03, 0.05)(0.3453, 0.6547)100.6895(0.03, 0.07)(0.4128, 0.5872)100.4825(0.05, 0.03)(0.2567, 0.7433)100.8528(0.05, 0.05)(0.3115, 0.6885)100.4599(0.05, 0.07)(0.3684, 0.6316)100.2800(0.07, 0.03)(0.2299, 0.7701)100.7201(0.07, 0.05)(0.2767, 0.7233)100.3495(0.07, 0.07)(0.3269, 0.6731)100.1908By analyzing Table4 and Figure 5, the following conclusions can be drawn:(1)
In the case of fixing the loss reduced by investing in one influencing factor, the expected loss of cities is negatively correlated with the loss reduced by investing in the other factor. That is, the greater the loss reduced by unit fund invested in influencing factors, the lower the expected loss from the cities. It implies that the influencing factor is less important to urban resource allocation and investment therein can be decreased slightly; on the contrary, the lower the loss reduced by unit fund invested in an influencing factor is, the larger the expected loss. This indicates that the influencing factor is of great importance to urban resource allocation and it is essential to increase the resource allocated thereto, provided the loss reduced per unit funding is low.(2)
On the condition of fixing the probabilities that the data and application software are invaded by illegal users and keepingv1<v2, the investment in the data is always low while in the application software is large in terms of the overall expectation of the resource allocation regardless of changes of the loss reduced by unit fund invested by cities in influencing factors.Figure 5
Changes in the expected loss withE1 and E2. In this figure, x axis denotes the losses reduced per unit funding invested in data (E1), y axis denotes the losses reduced per unit funding invested in application software (E2), and z axis denotes the expected loss of smart cities.


## 6. Conclusion

The relationship between various factors is analyzed based on factors influencing the information security of smart cities. On this basis, a four-hierarchy interpretative structural model for urban resources influencing information security is established. The data and application software in the first hierarchy of the structural model are explored. According to the actual condition, some assumptions are proposed to establish the corresponding evolutional game model; according to the actual condition in the first hierarchy of the interpretative structural model, the Nash equilibrium solutions when considering the influences of data and application systems are derived. Subsequently, the three types of typical problems in an actual situation are analyzed to verify the effectiveness of the structural model, and the correctness of its derivation is ascertained.Some future works can be summarized as follows: (1) more typical scenarios can be developed and verify the effectiveness of this proposed method, and (2) more recently published methods can be compared with the proposed method, and furtherly improved the proposed method.



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*Source: 6932163-2022-08-11.xml*