# A Study on Nash-Collative Differential Game ofN-Players for Counterterrorism

Authors: Mohamed Abd El-Hady Kassem; Abd El-Monem A. Megahed; Hebatallah K. Arafat
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this work, by using the Nash-collative approach for a differential game problem betweenN-governments and terrorist organizations, we study governments’ cooperation and the role of each government for counterterrorism. Furthermore, we discuss the intertemporal strategic interaction of governments and terrorist organizations, where all world governments have to cooperate to fight terrorism. Also, we study the necessary conditions for finding the optimal strategies for each government to fight terrorism; we discuss the existence of the solution of the formulated problem and the stability set of the first kind of the optimal strategies.


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## Body

## 1. Introduction

It is clear that the world suffers from many serious problems, and the problem of terrorism is one of the most important and serious problems suffered by the local and international communities throughout the ages, especially in recent times. Because of this, countries are controlled, their wealth is plundered, chaos and ignorance spread among peoples, the country’s political and religious identities are lost, individuals are destroyed intellectually, they become truly aimless, corruption prevails, and states fall. To face this problem and uproot terrorism from its roots, cooperation must be made between the governments of different countries and the different governments of the same state. This cooperation extends to the members of all societies, coordination occurs between the governments of different countries, and the public good must be upheld over personal interests. Governments have taken security measures to combat terrorism, such as freezing the assets of terrorist organizations and invading their territories to assassinate terrorists. The measures take into account the reactions of terrorists.The strength of a terrorist organization changes over time, as terrorists are recruited by existing terrorists, and the rate of terrorist recruitment is affected by their actions and the government’s antiterrorist actions. The strength of the organization is evaluated by its resources and activities, such as arming, funding, and the expertise of technology.The government derives its benefits from reducing terrorist resources and activities, in addition to demonstrating that these terrorist organizations are indiscriminate, but they incur costs through fighting terrorism. However, terrorist organizations try to maximize their power in terms of scale and terrorist attacks. Consequently, this study investigates how to help governments fight terrorism.In this research, we present and study this problem, explain the cooperation of governments with each other, and formulate this problem as a differential game between different governments and terrorist organizations. To clarify the cooperation of governments, to find solutions and to derive optimal strategies for combating terrorism, we will provide a Nash-collative approach to infer the necessary conditions for finding the optimal strategies to combat terrorism. Also, we study the existence of the solution, find it, and study its stability.The global reputation and optimal control of terrorism was discussed by Caulkins et al. [1]. They proved that success in fighting terrorism relies on community opinion, and the efficiency of water and fire strategies were studied by Caulkins et al. [2]. The first approach implementing a fuzzy differential game to guard territory was discussed by Hsia et al. [3, 4]. They considered the problem to guard territory as a differential game with fuzziness in the distance between the evader and the one guarding, and they discussed that the strategy for this problem is fuzzy. A parametric study of a Nash-collative differential game was discussed by Youness et al. [5]. A differential game as a large-scale problem was discussed by Youness et al. [6]. They presented a Nash approach to solve it. Furthermore, they studied Nash and min–max zero-sum approaches to get the optimal strategy of the differential game problem with fuzzy on the minimum of the objective function [7–9]. Nova et al. [10] studied the Stackelberg and Nash approaches of a differential game in addition to the sensitivity analysis. Cross-country strategic connectivity has been introduced to fight terrorism by Roy and Paul [11]. They analyzed the responses of equilibrium (in terms of defense, R&D, and preemption) to a possible terrorist strafe in a two-country framework using a multistep game with incomplete information. A min–max approach of a differential game was discussed to get the optimal solution of the government and the terror organization by Megahed [12–14]. He studied two problems of governments’ visions and terrorist organization and proved that governmental activities are important for fighting terrorism and discussed the Stackelberg differential game with E-differentiable function and E-convex set. In [15], Wrzaczek et al. discussed models of differential terror queue games, as terrorist organizations seek to increase the rates of attacks over time, but at the same time, the government is developing its antiterror activities; in [16], Megahed introduced the Stackelberg approach for counterterrorism.


## 2. Nash-Collative Differential Game

Definition 1.
A Nash-collative game is a Nash-equilibrium game in which some players make coalitions.


### 2.1.N+M Differential Games

Consider that we haveN+M players; the N players cost
(1)Jiu1,u2,⋯uN,v1,v2,⋯,vM=ϕixtf+∫t0tfe−ρtIit,xt,u1,u2,⋯uN,v1,v2,⋯,vMdt,i=1,2,⋯N,and theM players cost
(2)Jj′u1,u2,⋯uN,v1,v2,⋯,vM=ϕi′xtf+∫t0tfe−ηtIj′t,xt,u1,u2,⋯uN,v1,v2,⋯,vMdt,j=1,2,⋯M,subject to(3)ẋt=fxt,u1,u2,⋯uN,v1,v2,⋯,vM,xt0=x0,wherext∈Rn is the state trajectory of the game and xt0 is the initial state for all players.u1t∈Rs1,u2t∈Rs2,⋯,uNt∈RsN and v1t∈Rm1,v2t∈Rm2,⋯,vMt∈RmM denote the control or the decision of N and M players, respectively, which is taken to be a piecewise continuous function of time.f:t0,tf×Rn×Rm×Rs⟶Rn,Ii:t0,tf×Rn×Rm×Rs⟶Rn,Ij:t0,tf×Rn×Rm×Rs⟶Rn are C1; ϕixtf is the terminal payoff of the player i,i=1,2,.⋯,N, and ϕi′xtf is terminal payoff of the player j,j=1,2,.⋯M; and Iit,xt,u1,u2,⋯uN,v1,v2,⋯,vM is the running payoff of the player i,i=1,2,.⋯,N, and Ij′t,xt,u1,u2,⋯uN,v1,v2,⋯,vM is the running payoff of the player j,j=1,2,.⋯,M.Definition 2.
The admissible controlv∗=v1∗,⋯,vM∗ is said to be a Nash-collative optimal solution if and only if for all admissible v=v1,v2,⋯,vM, we have
(4)Jiu1∗,u2∗,⋯,uN∗,v1∗,v2∗,⋯,vM∗≤Jiu1,u2,⋯,uN,v1∗,v2∗,⋯,vM∗,i=1,2,⋯,N,Jj′u1∗,u2∗,⋯,uN∗,v1∗,v2∗,⋯,vM∗≤Jj′u1∗,u2∗,⋯,uN∗,v1,v2,⋯,vM,j=1,2,⋯,M.Definition 3.
Pareto optimal solution: let the controls v∗=v1∗,v2∗,⋯,vM∗ be admissible. If there exists w′∈RM, with wj′>0,j=1,2⋯,M, and ∑j=1Mwi′=1 and w∈RN,wi>0,i=1,2,⋯,N,∑i=1Nwi=1 such that for all v=v1,v2,⋯,vM, we have
(5)∑i=1NwiJiu1∗,u2∗,⋯,uN∗,v1∗,v2∗,⋯,vM∗≤∑i=1NwiJiu1,u2,⋯,uN,v1∗,v2∗,⋯,vM∗,∑j=1Mwi′Jj′u1∗,u2∗,⋯,uN∗,v1∗,v2∗,⋯,vM∗≤∑j=1Mwi′Jj′u1∗,u2∗,⋯,uN∗,v1,v2,⋯,vM.From the concept of coalitions, we suppose thatN of these players agree to form a coalition and play against the other players outside the coalition. Let u=u1,u2,⋯,uN∈Rs,s=∑i=1Nsi be the composite control for the players in the coalition and v=v1,v2,⋯,vM∈Rm,m=∑j=1Mmj be the composite control outside the coalition. The problem can be formulated as follows: find u∗,v∗ that solves the problems
(6)minuJu,v∗=Φxtf+∫t0tfe−ρtIxt,u,v∗dt,where
(7)I=∑i=1NwiIi,Φxtf=∑i=1NwiΦixtf,∑i=1Nwi=1,wi≥0,(8)minvJ′u∗,v=Φ′xtf+∫t0tfe−ηtIxt,u∗,vdt,subject to(9)ẋ=fxt,u∗,v,xt0=x0,where
(10)I′=∑j=1Mwj′Ij′,Φ′xtf=∑j=1Mwj′Φjxtf,∑j=1Mwj′=1,wj′≥0.Theorem 4.
Ifx∗,u∗,v∗ is an open-loop Nash-equilibrium solution for the problems (6), (8), and (9), then there exist continuous costate functions λt:t0,tf⟶Rn,qt:t0,tf⟶Rn and the Hamiltonian functions
(11)Hx,u,v∗,w1,⋯,wN,λ=Ix,u,v∗+λfx,u,v∗,H′x,u∗,v,w1′,⋯,wM′,qt=I′x,u∗,v+qtfx,u∗,v,whereu∗=u1∗,u2∗,⋯,uN∗ and v∗=v1∗,v2∗,⋯,vM∗ such that the following relations are satisfied:
(12)λ̇t=ρλ−∂Hx,u∗,v,w1,⋯,wNλ∂x,λtf=∑i=1Nwi∂ϕixtf∂x,(13)q̇t=ηqt−∂H′x,u,v∗,w1′,⋯,wM′,qt∂x,qtf=∑j=1Mwi′∂ϕi′xtf∂x,(14)Hx,u∗,v∗,w1,⋯,wN,λ≤Hx,u,v∗,w1,⋯,wN,λ,H′x,u∗,v∗,w1′,⋯,wM′,qt≤H′x,u∗,v,w1′,⋯,wM′,qt.For the proof, see [7].


## 2.1.N+M Differential Games

Consider that we haveN+M players; the N players cost
(1)Jiu1,u2,⋯uN,v1,v2,⋯,vM=ϕixtf+∫t0tfe−ρtIit,xt,u1,u2,⋯uN,v1,v2,⋯,vMdt,i=1,2,⋯N,and theM players cost
(2)Jj′u1,u2,⋯uN,v1,v2,⋯,vM=ϕi′xtf+∫t0tfe−ηtIj′t,xt,u1,u2,⋯uN,v1,v2,⋯,vMdt,j=1,2,⋯M,subject to(3)ẋt=fxt,u1,u2,⋯uN,v1,v2,⋯,vM,xt0=x0,wherext∈Rn is the state trajectory of the game and xt0 is the initial state for all players.u1t∈Rs1,u2t∈Rs2,⋯,uNt∈RsN and v1t∈Rm1,v2t∈Rm2,⋯,vMt∈RmM denote the control or the decision of N and M players, respectively, which is taken to be a piecewise continuous function of time.f:t0,tf×Rn×Rm×Rs⟶Rn,Ii:t0,tf×Rn×Rm×Rs⟶Rn,Ij:t0,tf×Rn×Rm×Rs⟶Rn are C1; ϕixtf is the terminal payoff of the player i,i=1,2,.⋯,N, and ϕi′xtf is terminal payoff of the player j,j=1,2,.⋯M; and Iit,xt,u1,u2,⋯uN,v1,v2,⋯,vM is the running payoff of the player i,i=1,2,.⋯,N, and Ij′t,xt,u1,u2,⋯uN,v1,v2,⋯,vM is the running payoff of the player j,j=1,2,.⋯,M.Definition 2.
The admissible controlv∗=v1∗,⋯,vM∗ is said to be a Nash-collative optimal solution if and only if for all admissible v=v1,v2,⋯,vM, we have
(4)Jiu1∗,u2∗,⋯,uN∗,v1∗,v2∗,⋯,vM∗≤Jiu1,u2,⋯,uN,v1∗,v2∗,⋯,vM∗,i=1,2,⋯,N,Jj′u1∗,u2∗,⋯,uN∗,v1∗,v2∗,⋯,vM∗≤Jj′u1∗,u2∗,⋯,uN∗,v1,v2,⋯,vM,j=1,2,⋯,M.Definition 3.
Pareto optimal solution: let the controls v∗=v1∗,v2∗,⋯,vM∗ be admissible. If there exists w′∈RM, with wj′>0,j=1,2⋯,M, and ∑j=1Mwi′=1 and w∈RN,wi>0,i=1,2,⋯,N,∑i=1Nwi=1 such that for all v=v1,v2,⋯,vM, we have
(5)∑i=1NwiJiu1∗,u2∗,⋯,uN∗,v1∗,v2∗,⋯,vM∗≤∑i=1NwiJiu1,u2,⋯,uN,v1∗,v2∗,⋯,vM∗,∑j=1Mwi′Jj′u1∗,u2∗,⋯,uN∗,v1∗,v2∗,⋯,vM∗≤∑j=1Mwi′Jj′u1∗,u2∗,⋯,uN∗,v1,v2,⋯,vM.From the concept of coalitions, we suppose thatN of these players agree to form a coalition and play against the other players outside the coalition. Let u=u1,u2,⋯,uN∈Rs,s=∑i=1Nsi be the composite control for the players in the coalition and v=v1,v2,⋯,vM∈Rm,m=∑j=1Mmj be the composite control outside the coalition. The problem can be formulated as follows: find u∗,v∗ that solves the problems
(6)minuJu,v∗=Φxtf+∫t0tfe−ρtIxt,u,v∗dt,where
(7)I=∑i=1NwiIi,Φxtf=∑i=1NwiΦixtf,∑i=1Nwi=1,wi≥0,(8)minvJ′u∗,v=Φ′xtf+∫t0tfe−ηtIxt,u∗,vdt,subject to(9)ẋ=fxt,u∗,v,xt0=x0,where
(10)I′=∑j=1Mwj′Ij′,Φ′xtf=∑j=1Mwj′Φjxtf,∑j=1Mwj′=1,wj′≥0.Theorem 4.
Ifx∗,u∗,v∗ is an open-loop Nash-equilibrium solution for the problems (6), (8), and (9), then there exist continuous costate functions λt:t0,tf⟶Rn,qt:t0,tf⟶Rn and the Hamiltonian functions
(11)Hx,u,v∗,w1,⋯,wN,λ=Ix,u,v∗+λfx,u,v∗,H′x,u∗,v,w1′,⋯,wM′,qt=I′x,u∗,v+qtfx,u∗,v,whereu∗=u1∗,u2∗,⋯,uN∗ and v∗=v1∗,v2∗,⋯,vM∗ such that the following relations are satisfied:
(12)λ̇t=ρλ−∂Hx,u∗,v,w1,⋯,wNλ∂x,λtf=∑i=1Nwi∂ϕixtf∂x,(13)q̇t=ηqt−∂H′x,u,v∗,w1′,⋯,wM′,qt∂x,qtf=∑j=1Mwi′∂ϕi′xtf∂x,(14)Hx,u∗,v∗,w1,⋯,wN,λ≤Hx,u,v∗,w1,⋯,wN,λ,H′x,u∗,v∗,w1′,⋯,wM′,qt≤H′x,u∗,v,w1′,⋯,wM′,qt.For the proof, see [7].


## 3. Stability

The problem is stable if it is persistent with regard to the data; that is, the problem is stable if when we change the problem “a little,” the solution changes only a little.


### 3.1. Stability with respect to the Parametersw and w′

We study the stability of the problem with respect to the weightsw and w′ for the Nash-collative differential game to show the role of each player in the coalition.Definition 5.
The stability Nash-collative differential game is denoted by(15)Bw,w′=Λ:Λ=w,w′∈RN+M,
such that the solution of the problem exists, wherew=w1,w2,⋯,wN,∑i=1Nwi=1, and w′=w1′,⋯,wM′,∑i=1Mwj′=1.Definition 6.
Suppose thatw,w′∈Bw,w′ with the corresponding the Nash-collative differential game; the stability set of the first kind for the Nash-collative differential game is defined by
(16)Su∗,v∗=Λ=w,w′∈RN+M∣u∗,v∗isasolutionoftheproblems6,7,8.Lemma 7.
If the cost functionalsJiu,v,w and Jj′u,v,w′(17)Ju,v,w=∑i=1NwiJiu1,⋯,uN,v1,⋯,vM,J′u,v,w′=∑j=1Mwj′Jj′u1,⋯,uN,v1,⋯,vM,
are linear with respect towi and wi′, respectively, then Su∗,v∗ is convex.Proof.
Suppose thatΛ1=w1,w1′,Λ2=w2,w2′∈S, then
(18)Ju∗,v∗,w1≤Ju,v∗,w1,J′u∗,v∗,w1′≤J′u,v∗,w1′,(19)Ju∗,v∗,w2≤Ju,v∗,w2,J′u∗,v∗,w2′≤J′u,v∗,w2′.
By multiplying both sides of the above inequalities byα and 1−α, respectively, and adding them, then
(20)Ju∗,v∗,αw1+1−αw2≤Ju,v∗,αw1+1−αw2,J′u∗,v∗,αw1′+1−αw2′≤J′u,v∗,αw1′+1−αw2′,
thenSu∗,v∗ is convex with respect to the weight parameters w and w′. ☐☐Remark 8.
If the cost functionalsJu,v,w,J′u,v,w′ are continuous on the weight space RN+M, where 0≤w≤1 and 0≤w′≤1, then the stability of the first kind Su∗,v∗ is closed.


## 3.1. Stability with respect to the Parametersw and w′

We study the stability of the problem with respect to the weightsw and w′ for the Nash-collative differential game to show the role of each player in the coalition.Definition 5.
The stability Nash-collative differential game is denoted by(15)Bw,w′=Λ:Λ=w,w′∈RN+M,
such that the solution of the problem exists, wherew=w1,w2,⋯,wN,∑i=1Nwi=1, and w′=w1′,⋯,wM′,∑i=1Mwj′=1.Definition 6.
Suppose thatw,w′∈Bw,w′ with the corresponding the Nash-collative differential game; the stability set of the first kind for the Nash-collative differential game is defined by
(16)Su∗,v∗=Λ=w,w′∈RN+M∣u∗,v∗isasolutionoftheproblems6,7,8.Lemma 7.
If the cost functionalsJiu,v,w and Jj′u,v,w′(17)Ju,v,w=∑i=1NwiJiu1,⋯,uN,v1,⋯,vM,J′u,v,w′=∑j=1Mwj′Jj′u1,⋯,uN,v1,⋯,vM,
are linear with respect towi and wi′, respectively, then Su∗,v∗ is convex.Proof.
Suppose thatΛ1=w1,w1′,Λ2=w2,w2′∈S, then
(18)Ju∗,v∗,w1≤Ju,v∗,w1,J′u∗,v∗,w1′≤J′u,v∗,w1′,(19)Ju∗,v∗,w2≤Ju,v∗,w2,J′u∗,v∗,w2′≤J′u,v∗,w2′.
By multiplying both sides of the above inequalities byα and 1−α, respectively, and adding them, then
(20)Ju∗,v∗,αw1+1−αw2≤Ju,v∗,αw1+1−αw2,J′u∗,v∗,αw1′+1−αw2′≤J′u,v∗,αw1′+1−αw2′,
thenSu∗,v∗ is convex with respect to the weight parameters w and w′. ☐☐Remark 8.
If the cost functionalsJu,v,w,J′u,v,w′ are continuous on the weight space RN+M, where 0≤w≤1 and 0≤w′≤1, then the stability of the first kind Su∗,v∗ is closed.


## 4. Counterterrorism Problem

In this section, we discuss the governments and terrorist problems and find the optimal strategies for the government to fight terrorism.


### 4.1. Governments’ Problem

Consider the problem ofN-governments and one terrorist organization where the cost functional of the government i, i=1,2,⋯,N, is
(21)maxuiJi=∫0∞e−ρitγihiui,v−cixit−kvt−αiuitdt,i=1,2,⋯,N,and the cost functional of the terrorist organization is(22)maxvJ0=∫0∞e−ρtσixit+βvdt,subject to(23)ẋi=rixit−hiuit,vt,wherext is the state trajectory of the game which is the stock of the terrorist organization which includes the financial capital, network supporters, and weapons. u is the strategy of the governments, v is strategy of the terrorist organization, hu,v is the interaction between the government and the terrorist organization (harvest function), and ρ is the positive decreasing rate of the running payoff. Since the governments are cooperative, then there exist wi∈Rs,wi≥0,∑i=1Nwi=1(24)J=∑i=1NwiJi=∫0∞∑i=1Ne−ρitwiγihiui,v−cixit−kvt−αiuit,(25)ẋi=rixit−hiui,v,suppose that the decreasing rate (ρi) are the same for all governments.Then, the Hamiltonian function of the Nash-collative problem(26)H=∑i=1Nwiγihiui,v−∑i=1Nwicixit−kvt−∑i=1Nαiwiui+λirixi−hiuit,vt.Since in counterterrorism, the governments have to maximize the Hamiltonian, we obtain the necessary conditions(27)∂H∂ui=wiγi−λi∂hiui,v∂ui−αiwi=0⇒ui∗=ui∗wi,λi,αi,v.Consider the harvest functionhiui,v=uiδvε,0<δ<ε<1(28)∂hi∂ui=δuiδ−1vε⇒ui=αiwiwiγiδ−δλi1/δ−1v−ε/δ−1.In (28), when terrorist organizations become active, governments double their activities in order to root out terrorism; this is evident in Figure 1.Figure 1
The relation between the strategies of governmentsui and the strategy of terrorism v.The costate variable has to follow the differential equation(29)λ̇i=ρλi−∂H∂xi=ρ+riλi−wici.With the optimal controlui∗v, the harvest function is
(30)hui,v=αiwiwiγiδ−δλiδ/δ−1v−ε/δ−1.


### 4.2. The Terrorist Organization Problem

Consider the payoff functional of the terrorist organization(31)maxvJ0=∫0∞e−ρtσixit+βvdt,and the state trajectory(32)ẋi=rixit−hiuit,vt.The Hamiltonian function of the terrorism(33)H0=σixit+βv+ηirixi−hiui,v.As the optimal control of the terrorists have to maximize the Hamiltonian function(34)∂H0∂v=β−ηi∂hiui,v∂v=0⇒v∗=v∗ui.The costate variable for the terrorist problem(35)η̇i=μηi−∂H0∂xi=μ−riηi−σi.


### 4.3. Existence

Now, we discuss the existence of the solution for the following system of differential equation; then, we have to integrate equations (36), (37), and (38) on the interval 0,t,
(36)ẋi=rixit−hiui,v,(37)λ̇i=ρ+riλi−wici,(38)η̇i=μ−riηi−σi,with the initial conditionsx0=x0,λi0=0, and ηi0=0, then
(39)xi=x0+∫0trixit−hiui,vdt,(40)λi=∫0tρ+riλidt−wicit,(41)ηi=∫0tμ−riηidt−σit.By differentiating the equations (39), (40), and (41), we get
(42)ẋi=rixit−hiui,v,λ̇i=ρ+riλi−wici,η̇i=μ−riηi−σi,puttingt=0 in the equations (39), (40), and (41); we get x0=x0,λi0=0, and ηi0=0, then the solution exists.Proposition 9.
The optimal control of the differential game problems (6), (9), and (8) with the initial conditions xi0=x0,λi0=0, and ηi0=0 are
(43)ui=αiwiwiγiδ−λiδε−1/1−δ−εβεηiε/ε+δ−1,v=βεηiδ−1/1−ε−δαiwiwiγiδ−δλiδ/1−ε−δ,
and the harvest function(44)hiui∗,v∗=αiwiwiγiδ−λiδδ2ε−1/1−δ−εβεηi1/ε+δ−1.Proof.
According to equations (27) and (28), we have
(45)ui=αiwiwiγiδ−δλi1/δ−1v−ε/δ−1.
Also, from (34) and ∂hu,v/∂v=εuiδvε−1, we have
(46)v=βεηi1/ε−1ui−δ/ε−1.
By solving equations (45) and (46), we get
(47)ui=αiwiwiγiδ−λiδε−1/1−δ−εβεηiε/ε+δ−1,v=βεηiδ−1/1−ε−δαiwiwiγiδ−δλiδ/1−ε−δ,
with the harvest function(48)hiui∗,v∗=αiwiwiγiδ−λiδδ2ε−1/1−δ−εβεηi1/ε+δ−1.
The solution of this problem is stable when the weight parameterwi is greater than λi/γi where λi is the costate vector of the government i and γi is the cost coefficient of the harvest function in the payoff of the government i. Also, the role of each government (wi), to fight terrorism, is greater than λi/γi. Thus, the stability set of the first kind for this problem is defined as
(49)Su∗,v∗=Λ=wi∈RN,wi>λiγi,
such that(50)ui=αiwiwiγiδ−λiδε−1/1−δ−εβεηiε/ε+δ−1,v=βεηiδ−1/1−ε−δαiwiwiγiδ−δλiδ/1−ε−δ,
is the optimal strategy of the counterterrorism problems (21), (22), and (23).
In (46), when governments play their role to the fullest, the government’s strategy reaches stability, and this is evident in the red and black curves, also, when the government gives up its role, the governments change their strategies and take strong measures to combat terrorism; this is evident in Figure 2.
In (47), when the government performs its role, the organizations increase their activities until they stabilize at a certain level, this is evident in the red, black, and blue curves, but when the government gives up its role, the terrorist organization is active at a high level, this is evident in the blue curve; this is evident in Figure 3.
In (48), when governments fulfill its role fully, the harvest function is greatest; this is evident in Figure 4.☐Proposition 10.
The state trajectory (the source stock of the terrorist) and the costate variablesλi and ηi of governments and terrorists are
(51)xit=xi0−hiui,vrierit+hiui,vri,λit=wiciρ+ri−wiciρ+rieρ+rit,ηit=σiμ−ri−σiμ−rieμ−rit.Proof.
Since the dynamic system of source stock is(52)ẋit=rixi−hui,v.
For the constant strategies and the harvest function, then(53)xite−rit=hiui,vrie−rit+constantc,
sincexit⟶xi0 as t⟶0, then c=xi0−hiui,v/ri and
(54)xit=xi0−hiui,vrierit+hiui,vri.
Since the costate variable for the government is(55)λ̇i=ρ+riλi−wici.
The solution of this differential equation(56)λie−ρ+rit=wiciρ+rie−ρ+rit+Cconstant,
sinceλit⟶0 as t⟶0, then C=−wici/ρ+ri and
(57)λit=wiciρ+ri−wiciρ+rieρ+rit.
Also, the costate variable of the terrorist is(58)η̇i=μ−riηi−σi.
Then,(59)ηie−μ−rit=+σμ−rie−μ−rit+cconstant,
sinceηi0=0, then c=−σi/μ−ri and
(60)ηit=σiμ−ri−σiμ−rieμ−rit.
☐☐Remark 11.
As shown in (54), the resource stock is increasing with time when the initial stock xi0 is greater than
(61)1riαiwiwiγiδ−λiδδ2ε−1/1−δ−εβεηi1/ε+δ−1.Remark 12.
The resource stockxit in the duration time 0,∞ is
(62)xit=1riαiwiwiγiδ−λiδδ2ε−1/1−δ−εβεηi1/ε+δ−1.Figure 2
The relation between the government’s strategiesui and the weight parameter wi.Figure 3
The relation between the strategy of terroristv and the weight parameter of each government wi.Figure 4
The relation between the harvest functionhui,v and weight parameters wi. ☐The resource stockxit directly decays when the role of each government wi is sufficiently larger.Lemma 13.
For the optimal strategiesui for the Nash-collative, v for the terrorist, and the harvest function hiui,v, the objective values of the cooperative governments J and the terrorist J0 are
(63)J=∑i=1NwiJi,(64)Ji=1ργihiui,v−kv−αiui−ciρhiui,v−1ρ−rixi0+hiui,vri,J0=σiμ−rixi0−hiui,vri+σiμrihui,v+βμv.Proof.
Since the objective function of governmenti is
(65)Ji=∫0∞e−ρitγihiui,v−cixit−kvt−αiuitdt,i=1,2,⋯,N,
and the state rejector is(66)xit=xi0−hiui,vrierit+hiui,vri.
By substituting from (66) in (65) and integrating it, we have
(67)Ji=1ργihiui,v−kv−αiui−ciρhiui,v−1ρ−rixi0+hiui,vri.
Also, by substituting from (66) in (68), we have
(68)J0=σiμ−rixi0−hiui,vri+σiμrihui,v+βμv,
whereui,v, and hiui,v are defined in Proposition 9. ☐☐


## 4.1. Governments’ Problem

Consider the problem ofN-governments and one terrorist organization where the cost functional of the government i, i=1,2,⋯,N, is
(21)maxuiJi=∫0∞e−ρitγihiui,v−cixit−kvt−αiuitdt,i=1,2,⋯,N,and the cost functional of the terrorist organization is(22)maxvJ0=∫0∞e−ρtσixit+βvdt,subject to(23)ẋi=rixit−hiuit,vt,wherext is the state trajectory of the game which is the stock of the terrorist organization which includes the financial capital, network supporters, and weapons. u is the strategy of the governments, v is strategy of the terrorist organization, hu,v is the interaction between the government and the terrorist organization (harvest function), and ρ is the positive decreasing rate of the running payoff. Since the governments are cooperative, then there exist wi∈Rs,wi≥0,∑i=1Nwi=1(24)J=∑i=1NwiJi=∫0∞∑i=1Ne−ρitwiγihiui,v−cixit−kvt−αiuit,(25)ẋi=rixit−hiui,v,suppose that the decreasing rate (ρi) are the same for all governments.Then, the Hamiltonian function of the Nash-collative problem(26)H=∑i=1Nwiγihiui,v−∑i=1Nwicixit−kvt−∑i=1Nαiwiui+λirixi−hiuit,vt.Since in counterterrorism, the governments have to maximize the Hamiltonian, we obtain the necessary conditions(27)∂H∂ui=wiγi−λi∂hiui,v∂ui−αiwi=0⇒ui∗=ui∗wi,λi,αi,v.Consider the harvest functionhiui,v=uiδvε,0<δ<ε<1(28)∂hi∂ui=δuiδ−1vε⇒ui=αiwiwiγiδ−δλi1/δ−1v−ε/δ−1.In (28), when terrorist organizations become active, governments double their activities in order to root out terrorism; this is evident in Figure 1.Figure 1
The relation between the strategies of governmentsui and the strategy of terrorism v.The costate variable has to follow the differential equation(29)λ̇i=ρλi−∂H∂xi=ρ+riλi−wici.With the optimal controlui∗v, the harvest function is
(30)hui,v=αiwiwiγiδ−δλiδ/δ−1v−ε/δ−1.


## 4.2. The Terrorist Organization Problem

Consider the payoff functional of the terrorist organization(31)maxvJ0=∫0∞e−ρtσixit+βvdt,and the state trajectory(32)ẋi=rixit−hiuit,vt.The Hamiltonian function of the terrorism(33)H0=σixit+βv+ηirixi−hiui,v.As the optimal control of the terrorists have to maximize the Hamiltonian function(34)∂H0∂v=β−ηi∂hiui,v∂v=0⇒v∗=v∗ui.The costate variable for the terrorist problem(35)η̇i=μηi−∂H0∂xi=μ−riηi−σi.


## 4.3. Existence

Now, we discuss the existence of the solution for the following system of differential equation; then, we have to integrate equations (36), (37), and (38) on the interval 0,t,
(36)ẋi=rixit−hiui,v,(37)λ̇i=ρ+riλi−wici,(38)η̇i=μ−riηi−σi,with the initial conditionsx0=x0,λi0=0, and ηi0=0, then
(39)xi=x0+∫0trixit−hiui,vdt,(40)λi=∫0tρ+riλidt−wicit,(41)ηi=∫0tμ−riηidt−σit.By differentiating the equations (39), (40), and (41), we get
(42)ẋi=rixit−hiui,v,λ̇i=ρ+riλi−wici,η̇i=μ−riηi−σi,puttingt=0 in the equations (39), (40), and (41); we get x0=x0,λi0=0, and ηi0=0, then the solution exists.Proposition 9.
The optimal control of the differential game problems (6), (9), and (8) with the initial conditions xi0=x0,λi0=0, and ηi0=0 are
(43)ui=αiwiwiγiδ−λiδε−1/1−δ−εβεηiε/ε+δ−1,v=βεηiδ−1/1−ε−δαiwiwiγiδ−δλiδ/1−ε−δ,
and the harvest function(44)hiui∗,v∗=αiwiwiγiδ−λiδδ2ε−1/1−δ−εβεηi1/ε+δ−1.Proof.
According to equations (27) and (28), we have
(45)ui=αiwiwiγiδ−δλi1/δ−1v−ε/δ−1.
Also, from (34) and ∂hu,v/∂v=εuiδvε−1, we have
(46)v=βεηi1/ε−1ui−δ/ε−1.
By solving equations (45) and (46), we get
(47)ui=αiwiwiγiδ−λiδε−1/1−δ−εβεηiε/ε+δ−1,v=βεηiδ−1/1−ε−δαiwiwiγiδ−δλiδ/1−ε−δ,
with the harvest function(48)hiui∗,v∗=αiwiwiγiδ−λiδδ2ε−1/1−δ−εβεηi1/ε+δ−1.
The solution of this problem is stable when the weight parameterwi is greater than λi/γi where λi is the costate vector of the government i and γi is the cost coefficient of the harvest function in the payoff of the government i. Also, the role of each government (wi), to fight terrorism, is greater than λi/γi. Thus, the stability set of the first kind for this problem is defined as
(49)Su∗,v∗=Λ=wi∈RN,wi>λiγi,
such that(50)ui=αiwiwiγiδ−λiδε−1/1−δ−εβεηiε/ε+δ−1,v=βεηiδ−1/1−ε−δαiwiwiγiδ−δλiδ/1−ε−δ,
is the optimal strategy of the counterterrorism problems (21), (22), and (23).
In (46), when governments play their role to the fullest, the government’s strategy reaches stability, and this is evident in the red and black curves, also, when the government gives up its role, the governments change their strategies and take strong measures to combat terrorism; this is evident in Figure 2.
In (47), when the government performs its role, the organizations increase their activities until they stabilize at a certain level, this is evident in the red, black, and blue curves, but when the government gives up its role, the terrorist organization is active at a high level, this is evident in the blue curve; this is evident in Figure 3.
In (48), when governments fulfill its role fully, the harvest function is greatest; this is evident in Figure 4.☐Proposition 10.
The state trajectory (the source stock of the terrorist) and the costate variablesλi and ηi of governments and terrorists are
(51)xit=xi0−hiui,vrierit+hiui,vri,λit=wiciρ+ri−wiciρ+rieρ+rit,ηit=σiμ−ri−σiμ−rieμ−rit.Proof.
Since the dynamic system of source stock is(52)ẋit=rixi−hui,v.
For the constant strategies and the harvest function, then(53)xite−rit=hiui,vrie−rit+constantc,
sincexit⟶xi0 as t⟶0, then c=xi0−hiui,v/ri and
(54)xit=xi0−hiui,vrierit+hiui,vri.
Since the costate variable for the government is(55)λ̇i=ρ+riλi−wici.
The solution of this differential equation(56)λie−ρ+rit=wiciρ+rie−ρ+rit+Cconstant,
sinceλit⟶0 as t⟶0, then C=−wici/ρ+ri and
(57)λit=wiciρ+ri−wiciρ+rieρ+rit.
Also, the costate variable of the terrorist is(58)η̇i=μ−riηi−σi.
Then,(59)ηie−μ−rit=+σμ−rie−μ−rit+cconstant,
sinceηi0=0, then c=−σi/μ−ri and
(60)ηit=σiμ−ri−σiμ−rieμ−rit.
☐☐Remark 11.
As shown in (54), the resource stock is increasing with time when the initial stock xi0 is greater than
(61)1riαiwiwiγiδ−λiδδ2ε−1/1−δ−εβεηi1/ε+δ−1.Remark 12.
The resource stockxit in the duration time 0,∞ is
(62)xit=1riαiwiwiγiδ−λiδδ2ε−1/1−δ−εβεηi1/ε+δ−1.Figure 2
The relation between the government’s strategiesui and the weight parameter wi.Figure 3
The relation between the strategy of terroristv and the weight parameter of each government wi.Figure 4
The relation between the harvest functionhui,v and weight parameters wi. ☐The resource stockxit directly decays when the role of each government wi is sufficiently larger.Lemma 13.
For the optimal strategiesui for the Nash-collative, v for the terrorist, and the harvest function hiui,v, the objective values of the cooperative governments J and the terrorist J0 are
(63)J=∑i=1NwiJi,(64)Ji=1ργihiui,v−kv−αiui−ciρhiui,v−1ρ−rixi0+hiui,vri,J0=σiμ−rixi0−hiui,vri+σiμrihui,v+βμv.Proof.
Since the objective function of governmenti is
(65)Ji=∫0∞e−ρitγihiui,v−cixit−kvt−αiuitdt,i=1,2,⋯,N,
and the state rejector is(66)xit=xi0−hiui,vrierit+hiui,vri.
By substituting from (66) in (65) and integrating it, we have
(67)Ji=1ργihiui,v−kv−αiui−ciρhiui,v−1ρ−rixi0+hiui,vri.
Also, by substituting from (66) in (68), we have
(68)J0=σiμ−rixi0−hiui,vri+σiμrihui,v+βμv,
whereui,v, and hiui,v are defined in Proposition 9. ☐☐


## 5. Conclusions

In this intervention, the cooperation of governments is studied for fighting terrorism by using the Nash-collative approach; the necessity for finding the optimal strategies is derived and we proved that the solution exists and found it. Also, we showed that the optimal strategies are stable when the weight parameterwi is greater than λi/γi. Finally, we derived the objective value of each government and its role wi in counterterrorism, the objective of the terrorist organization, and its stock recourse.



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*Source: 7513406-2021-08-24.xml*