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Neural Computing and Applications

, Volume 30, Issue 3, pp 865–870 | Cite as

A differential game related to terrorism: min-max zero-sum two persons differential game

  • Abd El-Monem A. Megahed
Original Article

Abstract

In this work, we are going to study a differential game related to terrorism: min-max two persons differential game, the question and discuss the qualitative of how best to prosecute the “war on terror” leads to strategic interaction in an inter-temporal setting. We consider a min-max differential game between a governments and a terrorist organization. To obtain the optimal strategy of this problem, we study the analytic form of min-max two persons differential game and a necessary conditions of this problem. Furthermore, we study a saddle point of a min-max differential game. Finally, we make a comparison between the game of the government and the terrorist organization. In the final, I hope from among this study introduce the optimal control and state trajectory to the governments to counter-terrorist.

Keywords

Game theory Terrorism Differential game Min-max game Saddle points 

1 Introduction

The terrorism problems have been become very large and dangerous about the world, and we are using the operations research as one subject of mathematics for getting methods to combat the terrorism. Counter-terror measure range comes from security arrangements and the government’s activities to freeze the property of terrorist organizations and the invasion of their land. Any action for counter-terror, the government should be take into account the reaction of terrorists organizations. In the following, we are using the approach of a differential game to present the inter-temporal strategic interaction of government and terrorists organizations.

The organization’s power comes from the terrorists activities, weapons, financial capital, and technological expertise. Caulkins et al. [1] proposed that combating terrorism rely on the community opinion and Caulkins et al. [2] introduced the comparison between the efficiency of water and fair strategies.

The organization’s force changes with the time. The decreasing rate of terrorists dependent on their own actions and the government’s activities anti them. The government’s objectives derive benefit from losing resources for terrorists, but the fight against terrorism bears costs and disutility from the terrorist organization, the later tries to achieve maximum strength both in terms of size and terrorists attacks.

For all the above, we are going to discuss how to help the governments to anti-terrorist, a min-max differential game play the main role to fight terrorism.

A fuzzy differential game approach of guarding territory and a movable territory are introduced in Hsia and Hsie [3] and Hung et al. [4], a parametric Nash collative differential game is presented in Youness et al. [5], a study on fuzzy differential game, a study on large scale continuous differential game, min-max fuzzy continuous differential game and min-max continuous differential with fuzzy control are presented in Youness and Megahed [6, 7], Hegazy et al. [8] and Megahed and Hegazy [9].

A differential game related to terrorism: Nash and Stackelberg strategies are introduced in Nova et al. [10], terrorism deterrence in a two countries framework: strategic interactions between R & D, defense and preemption are presented in Roy and Paul [11], a global reputation and the optimal control of terrorism are presented in Caulkins et al. [1], and Caulkins et al. [2] discussed the water or fire strategies as an optimizing operations for combating terror.

2 Problem formulation

Let’s look at the differential game with the state trajectory y(t), that describe the resources of the International Terrorism Organization (ITO). It may include financial capital, weapons, and a supporter’s network, etc., \(t\in \left[ 0,\infty \right) \) is the time. The two players are the government with nonnegative strategy \(w_1(t)\) and the other side ITO with nonnegative strategy \(w_2(t)\) as the opponent. The stock of resources of ITO grows according to the growth of a linear function G(y), i.e., \(G(y)=ry,r>0.\) The attacks reduce the growth of the inventory of resources because it decreasing the number of terrorists (e.g., due to a suicide bombing or capture terrorists and kill them) in addition to the weapons and methods of funding, and it may include the limitation of supporter’s network. The decreasing of the growth resources dependent on the size and the number of attacks \(w_{2}(t)\) and the combating terrorism operations \(w_1(t).\) This influence is defined as the harvest function \(h(w_{1}{(t)},w_2{(t)}).\) The dynamic sequence of the resource inventory y(t) can be written as
$$ y^{\cdot }=r y-h(w_{1},w_{2}),\quad y(0)=y_{0}>0, $$
(1)
where \(y_{0}\) is the initial inventory of measures terrorists and the state variable is nonnegative
$$ y(t)\ge 0, \quad t\ge 0. $$
(2)
The increasing of anti-terrorism and carry out attacks makes a reduction of the resources growth, for this, we suppose that
$$ h_{w_{1}}(w_{1},w_{2})>0,\quad h_{w_{2}}(w_{1},w_{2})>0. $$
Anti-terror actions make a marginal decreasing of the efficient \(h_{w_{1} w_{1}}<0.\) Furthermore, the increasing of attacks make disproportionate larger losses of resources, i.e., \(h_{w_{2}w_{2}}>0.\) In the final, the tools that enhance each other, i.e., \(h_{w_{1}w_{2}}>0.\) This reaction shows that the marginal efficiency of the fight against terrorism increases with the terrorist attacks and can be controlled on the visible terrorist more than hidden ones. Additionally, we suppose that the Inada conditions in the economic literature are fulfilled
$$\begin{aligned} \lim _{w_{1}\rightarrow 0}h_{w_{1}}{(w_{1},w_{2})}= & \infty \quad \lim _{w_{1}\rightarrow \infty }h_{w_{1}}(w_{1},w_{2})=0 \\ \lim _{w_{2}\rightarrow 0}h_{w_{2}}(w_{1},w_{2})= & 0,\quad \lim _{w_{2}\rightarrow \infty }h_{w_{2}}(w_{1},w_{2})=\infty . \end{aligned}$$
These lead to the optimal controls are nonnegative, \(w_{1}(t)\ge 0\) and \(w_{2}(t)\ge 0,\) \(t>0.\)
Player 1 (Government): the utility of the player 1 is based on the loss of the terrorists’ stock but the disutility from the size of ITO, terrorists activities and the cost of anti-terror measures. For simplicity, all these terms are assumed to be linear. Then the target of the player 1
$$ \max _{{w_{1}(t)}}\left\{ J_{0}=\int _{0}^{\infty }e^{-\eta _{1}t}\left[ c_{1} h(w_{1},w_{2})-c_{2}y(t)-c_{3}w_{2}(t)-\alpha w_{1}(t)\right] {\mathrm {d}}t\right\} $$
(3)
where \(c_{1}>0,c_{2}>0,c_{3}>0\) and \(\alpha >0.\)
Player 2 (ITO) takes his utility from the resources inventory y(t) and the terrorists actions with the strategy \(w_{2}(t),\) and disutility from government activities. It is maximize the following problem
$$ \max _{w_{2}(t)} \left\{ J_{1}=\int _{0}^{\infty }e^{-\eta _{2}t}\left[ \mu _{1} y(t)+\mu _{2} w_{2}(t)-\frac{\rho }{u(t)}\right] {\mathrm {d}}t\right\} $$
(4)
where \(\mu _{1}>0,\) \(\mu _{2}>0 \) and \(\rho >0.\)
The decreasing rates \(\eta _{i},=1,2\) are assumed to be greater than the growth rate, r, i.e.,
$$ \eta _{i}>r, \,\,i=1,2. $$
(5)
In this paper, we calculate a min-max equilibria. The solution satisfies the Pontryagin’s maximum [9].

3 Min–max equilibrium

A min-max game is called antagonistic game for two persons (two players). In this paper, the player 1 is the government and the player 2 is the International Terror Organization (ITO). There are two cases for studying this problem:

3.1 The game of view’s of the government

In this case, the government is going to find the strategic variable \(w_{1}(t)\) to maximize his payoff and it is called the maximizing player, but the ITO tries to find the strategic variable \(w_{2}(t)\) to minimize this payoff and it is called the minimizing player, the game takes the following form
$$\begin{aligned} \left. \begin{array}{c} \min _{w_{2} (t)}\max _{w_{1}(t)}J_{0}=\int _{0}^{\infty }e^{-\eta _{1}t}\left[ c_{1}h(w_{1}(t),w_{2}(t))-c_{2}y(t)-c_{3}w_{2} (t)-\alpha w_{1}(t)\right] {\mathrm {d}}t \\ \\ y^{\cdot }=r y(t)-h(w_{1}(t),w_{2}(t)),\ \ \ \ \ \ y(0)=y_{0}>0,y(t)\ge 0 \, {\text{for \,all}}\,t \end{array} \right\} \end{aligned}$$
(6)
Note: we denote that
$$ I_{1}(\eta _{1},y(t),w_{1}(t),w_{2}(t))=c_{1}h(w_{1}(t),w_{2}(t))-c_{2}y(t)-c_{3}w_{2} (t)-\alpha w_{1}, $$
and
$$ f(y,w_{1},w_{2})=ry(t)-h(w_{1}(t),w_{2}(t)). $$

Definition 1

The point \((w_{1}^{*},w_{2}^{*})\) is said to be saddle point of the min-max continuous differential game problem 6 if
$$ J_{0}(w_{1}^{*},w_{2} ) \,\le \,J_{0}(w_{1}^{*},w_{2} ^{*}) \,\le\, J_{0}(w_{1},w_{2} ^{*}). $$
(7)

3.2 The necessary conditions of an open saddle point solution

Theorem 1

Let \(I_{1}(\eta _{1},y(t),w_{1}(t),w_{2}(t))\) and \(f(y,w_{1},w_{2})\) are continuous differentiable functions. If \((w_{1}^{*},w_{2}^{*})\) is saddle point with the state trajectory \(y^{*}(t)\) for the problem (Government). Then there exists a costate vector \(\lambda _{1}(t)\) and the Hamiltonian function \(H_{1}\) defined by
$$ H_{1}(y(t),w_{1}(t),w_{2}(t),\lambda (t))=I_{1}(\eta _{1},y(t),w_{1}(t),w_{2}(t))+\lambda _{1}(t)f(y,w_{1},w_{2}) $$
such that the following conditions are satisfied
$$\begin{aligned} \left. \begin{array}{c} \begin{array}{c} \frac{\partial H_{1}}{\partial w_{1}}=0,\ \ \ \ \ \ \frac{\partial H_{1}}{ \partial w_{2}}=0\ \\ \frac{\partial ^{2}H_{1}}{\partial w_{1}^{2}}\frac{\partial ^{2}H_{1}}{\partial w_{2}^{2}}-\left( \frac{\partial ^{2}H_{1}}{\partial w_{1}\partial w_{2}}\right) ^{2}\le 0,\ \ \ \ \ \frac{\partial ^{2}H_{1}}{\partial w_{1}^{2}}\le 0\ \ \ \ , \frac{\partial ^{2}H_{1}}{\partial w_{2}^{2}}\ge 0 \end{array}\\ \lambda _{1}^{\cdot }=\eta _{1}\lambda _{1}-\frac{\partial H_{1}}{\partial y}\\ \min _{w_{2}(t)} H_{1}(y(t),w_{1}^{*}(t),w_{2}(t),\lambda (t))=H_{1}(y(t),w_{1}^{*}(t),w_{2}^{*}(t),\lambda (t))=\max _{w_{1}(t)}H_{1}(y(t),w_{1}(t),w_{2}^{*}(t),\lambda (t)) \end{array} \right\} \end{aligned}$$
(8)

Proof

The proof of this theorem is similar to [9, Theorem 3.1]. \(\square \)

Since the optimal strategy of the government and ITO have to maximize and minimize the Hamiltonian function \(H_{1},\)
$$\begin{aligned} \left. \begin{array}{l} \frac{\partial H_{1}}{\partial w_{1}}=(c_{1}-\lambda _{1})h_{w_{1}}-\alpha =0\Longrightarrow h_{w_{1}}=\frac{\alpha }{c_{1}-\lambda _{1}} \\ \frac{\partial H_{1}}{\partial w_{2}}=(c_{1}-\lambda _{1})h_{w_{2}}-c_{3}=0\Longrightarrow h_{w_{2}}=\frac{c_{3}}{c_{1}-\lambda _{1}} \end{array} \right\} \end{aligned}$$
(9)
the adjoint variable satisfy the differential equation
$$ \lambda _{1}^{\cdot }=\eta _{1}\lambda _{1}-\frac{\partial H_{1}}{\partial y} =\lambda _{1}(\eta _{1}-r)+c_{2}, $$
(10)
and the transversality conditions
$$ \lim _{t\rightarrow \infty }e^{-\eta _{1}t}y(t)\lambda _{1}(t)=0 $$
(11)
then the solution of the adjoint equation is
$$ \lambda _{1}(t)=\left(\lambda _{0}+\frac{c_{2}}{(\eta _{1}-r)}\right)e^{(\eta _{1}-r)t}- \frac{c_2}{\eta _{1}-r}, $$
(12)
where \(\lambda _{1}(0)=\lambda _{0}.\) Since \(\eta _{1}>r,\) then \(\lambda _{1}(t)\rightarrow \infty \) as \(t\rightarrow \infty \) which is violating the transversality conditions except when choosing the constant steady state value
$$ \lambda _{1}=\lambda _{0}=-\frac{c}{\eta _{1}-r} $$
The Hamiltonian \(H_{1}\) is convex with respect to the strategy \(w_{2}\) and concave with respect to the strategy \(w_{1}\), and therefore, we find the maximization of \(H_{1}\) with respect to \(w_{1}\) and the minimization of \(H_{1}\) with respect to \(w_{2}\).

Consider the harvest function \(h(w_{1},w_{2})=w_{1}^{\tau }w_{2}^{\delta },\) with \(0\,<\,\tau\,<\,1\,<\,\delta \)

Proposition 1

The optimal strategies of the game 6 are given by
$$\begin{aligned} \left. \begin{array}{c} w_{1}=\left[ \left( \frac{\alpha }{\tau (c_{1}-\lambda _{1})}\right) ^{\delta-1}\left( \frac{c_{3}}{\delta (c_{1}-\lambda _{1})}\right) ^{-\delta} \right] ^{\frac{1}{1-\tau-\delta }} \\ w_{2}=\left[ \left( \frac{c_{3}}{\delta (c_{3}-\lambda _{1})}\right) ^{\tau-1}\left( \frac{\alpha }{\tau (c_{1}-\lambda _{1})}\right) ^{-\tau}\right] ^{ \frac{1}{1-\tau -\delta }} \end{array} \right\} \end{aligned}$$
(13)
with the harvest function
$$ h(w_{1},w_{2})=\left[ \left( \frac{\alpha}{\tau (c_{1}-\lambda _{1})}\right) ^{-\tau} \left( \frac{c_{3}}{\delta(c_{1}-\lambda _{1})}\right) ^{-\delta} \right] ^{\frac{1}{1-\tau-\delta }}$$
(14)

Proof

From the necessary conditions, \(\frac{\partial H_{1}}{\partial w_{1}}=0,\)  \(\frac{\partial H_{1}}{\partial w_{2}}=0,\) we have
$$ h_{w_{1}}= \tau w_{1}^{\tau -1}w_{2}^{\delta }=\frac{\alpha }{(c_{1}-\lambda _{1})},\,\hbox {then} \,\,w_{1}=\left( \frac{\alpha }{\tau (c_{1}-\lambda _{1})}\right) ^{\frac{1}{\tau-1}}w_{2}^{\frac{-\delta}{\tau-1 }} $$
(15)
and
$$ h_{w_{2}}= \delta w_{1}^{\tau }w_{2}^{\delta -1}=\frac{c_{3}}{ c_{1}-\lambda _{1}},\,\hbox {then} \,\,w_{2}=\left( \frac{c_{3}}{\delta (c_{1}-\lambda _{1})}\right) ^{\frac{1}{\delta -1}}w_{1}^{\frac{-\tau}{\delta -1 }} $$
(16)
and thus
$$\begin{aligned} \left. \begin{array}{c} \begin{array}{c} w_{1}=\left[ \left( \frac{\alpha }{\tau (c_{1}-\lambda _{1})}\right) ^{\delta-1}\left( \frac{c_{3}}{\delta (c_{1}-\lambda _{1})}\right) ^{-\delta} \right] ^{\frac{1}{1-\tau -\delta }} \\ w_{2}=\left[ \left( \frac{c_{3}}{\delta (c_{1}-\lambda _{1})}\right) ^{\tau-1}\left( \frac{\alpha }{\tau (c_{1}-\lambda _{1})}\right) ^{-\tau}\right] ^{ \frac{1}{1-\tau-\delta }} \end{array} \\ h(w_{1},w_{2})=\left[ \left( \frac{\alpha}{\tau ({c_1}-\lambda _{1})}\right) ^{-\tau} \left( \frac{c_{3}}{\delta({c_1}-\lambda _{1})}\right) ^{-\delta} \right] ^{\frac{1}{1-\tau-\delta }}\end{array} \right\} \end{aligned}$$
(17)
and
$$\begin{aligned} \left. \begin{array}{c} \left| \begin{array}{cc} H_{1w_{!}w_{2}} & H_{1w_{1}w_{2}} \\ \\ H_{1w_{2}w_{1}} & H_{1w_{2}w_{2}} \end{array} \right| =\left| \begin{array}{cc} \tau (\tau -1)w_{1}^{\tau -2}w_{2}^{\delta } & \tau \delta w_{1}^{\tau -1}w_{2}^{\delta -1} \\ \tau \delta w_{1}^{\tau -1}w_{2}^{\delta -1} & \delta (\delta -1)w_{1}^{\tau }w_{2}^{\delta -2} \end{array} \right| \\ =\tau \delta (1-\tau -\delta )w_{1}^{2(\tau -1)}w_{2}^{2(\delta -1)}<0 \end{array} \right\} \end{aligned}$$
(18)
i.e., \((w_{1},w_{2})\) is saddle point of the problem 6. \(\square \)

Lemma 1

The objective of the government (player 1) for the constant strategies \(w_{1},w_{2}\) is
$$ \mathrm {J}_{0}\mathrm {=}\frac{h}{\eta _{1}}\left( c_{1}+\frac{c_{2}}{\eta _{1}-r} \right) \mathrm {-}\frac{\alpha w_{1}}{\eta _{1}}\mathrm {-}\frac{c_{3}w_{2}}{\eta _{1}} \mathrm {-}\frac{c_{2}y_{0}}{\eta _{1}-r}. $$
(19)

Proof

The solution of the ordinary differential equation
$$ y^{\cdot }=r\ y(t)-h(w_{1}(t),w_{2}(t)) $$
is
$$ \begin{aligned} y(t)e^{-rt}=\frac{1}{r}e^{-rt}h(w_{1},w_{2})+c,\quad \\ \hbox {where} \,c \hbox { is \,constant} \end{aligned}$$
for \(t\rightarrow 0,\)  \(c=y_{0}-\frac{1}{r}h(w_{1},w_{2}),\) then
$$ y(t)=(y_{0}-\frac{1}{r}h(w_{1},w_{2}))e^{rt}+\frac{h}{r}, $$
(20)
and thus
$$ J_{0}=\frac{h}{\eta _{1}}(c_{1}-\frac{c_{2}}{r-\eta _{1}})-\frac{c_{3} w_{2}}{\eta _{1}}- \frac{\alpha w_{1}}{\eta _{1}}+\frac{c_{2}y_{0}}{r-\eta _{1}}, $$
(21)
\(\square \)

3.3 The game of view’s of ITO

In this case, the ITO is going to find the strategic variable \(w_{2}(t)\) to maximize his payoff and it is called the maximizing player, but the government tries to find the strategic variable \(w_{1}(t)\) to minimize his payoff and it is called the minimizing player, the game takes the following form
$$\begin{aligned} \left. \begin{array}{c} \min _{w_{1}(t)}\max _{w_{2} (t)}J_{1}= \int _{0}^{\infty }e^{-\eta _{2}t}\left[ \mu _{1} y(t)+\mu _{2} w_{2} (t)-\frac{\rho }{w_{1}(t)}\right] dt \\ y^{\cdot }=r y(t)-h(w_{1}(t),w_{2}(t)),\ \ \ \ \ \ \ \ \ \ \ \ y(0)=x_{0}>0, y(t)\ge 0 \,\hbox {for\,all}\,t \end{array} \right\} \end{aligned}$$
(22)
Note: we denote that \(\ I_{2}(\eta _{2},y(t),w_{1}(t),w_{2}(t))=\mu _{1} y(t)+\mu _{2} w_{2} (t)-\frac{\rho }{w_{1} (t)}\)

Definition 2

The point \((w_{1}^{*},w_{2}^{*})\) is said to be saddle point of the min-max continuous differential game problem 23 if
$$ J_{2}(w_{1}^{*},w_{2} ) \,\le \,J_{2}(w_{1}^{*},w_{2} ^{*})\le J_{2}(w_{1}~,w_{2} ^{*}). $$
(23)

Theorem 2

Let \(I_{2}(\eta _{2},y(t),w_{1}(t),w_{2}(t))\) and \(f(y,w_{1},w_{2})\) are continuous differentiable functions. If \((w_{1}^{*},w_{2}^{*})\) is saddle point with the state trajectory \(y^{*}(t)\) for the problem (ITO). Then there exists a costate vector \(\lambda _{2}(t)\) and the Hamiltonian function \(H_{2}\) defined by
$$ H_{2}(y(t),w_{1}(t),w_{2}(t),\lambda _{2}(t))=I_{2}(\eta _{2},y(t),w_{1}(t),w_{2}(t))\,+\,\lambda _{2}(t)f(y,w_{1},w_{2}) $$
such that the following conditions are satisfied
$$\begin{aligned} \left. \begin{array}{c} \begin{array}{c} \frac{\partial H_{2}}{\partial w_{1}}=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{ \partial H_{2}}{\partial w_{2}}=0\ \ \\ \\ \frac{\partial ^{2}H_{2}}{\partial w_{1}^{2}}\frac{\partial ^{2}H_{2}}{\partial w_{2}^{2}}-\left( \frac{\partial ^{2}H_{2}}{\partial w_{1}\partial w_{2}}\right) ^{2}\le 0,\ \frac{\partial ^{2}H_{2}}{\partial w_{1}^{2}}\ge 0\ \ \ \ ,\frac{ \partial ^{2}H_{2}}{\partial w_{2}^{2}}\le 0 \\ \\ \lambda _{2}^{\cdot }=\eta _{2}\lambda _{2}-\frac{\partial H_{2}}{\partial y} \end{array} \\ \max _{w_{2}(t)}\ \ H_{2}(y(t),w_{1}^{*}(t),w_{2}(t),\lambda _{2} (t))=H_{2}(y(t),w_{1}^{*}(t),w_{2}^{*}(t),\lambda _{2} (t))=\min _{w_{1}(t)}\ \ H_{2}(y(t),w_{1}(t),w_{2}^{*}(t),\lambda _{2} (t)) \end{array} \right\} \end{aligned}$$
(24)

Proof

The proof of this theorem is similar to [9, Theorem 3.1]. \(\square \)

Since the optimal strategy of the ITO and the government have to maximize and minimize the Hamiltonian function \(H_{2}\) respectively,
$$\begin{aligned} \left. \begin{array}{c} \frac{\partial H_{2}}{\partial w_{1}}=\frac{\rho }{w_{1}^{2}(t)}-\lambda _{2}h_{w_{1}}=0\Longrightarrow h_{w_{1}}=\frac{\rho }{\lambda _{2}w_{1}^{2}(t)} \\ \frac{\partial H_{2}}{\partial w_{2}}=\mu _{2} -\lambda _{2}h_{w_{2}}=0\Longrightarrow h_{w_{2}}=\frac{\mu _{2} }{\lambda _{2}} \end{array} \right\} \end{aligned}$$
(25)
the adjoint variable satisfy the differential equation
$$ \lambda _{2}^{\cdot }=\eta _{2}\lambda _{2}-\frac{\partial H_{2}}{\partial y} =\lambda _{2}(\eta _{2}-r)-\mu _{1}, $$
(26)
and the transversality conditions
$$ \lim _{t\rightarrow \infty }e^{-\eta _{2}t}y(t)\lambda _{2}(t)=0, $$
then the solution of the adjoint equation is
$$ \lambda _{2}(t)=\left(\lambda _{20}+\frac{\mu _{1} }{(\eta _{2}-r)}\right)e^{(\eta _{2}-r)t}-\frac{\mu _{1} }{\eta _{2}-r}, $$
(27)
where \(\lambda _{2}(0)=\lambda _{20}.\)

The Hamiltonian \(H_{2}\) is concave with respect to the strategy \(w_{2}\) and convex with respect to the strategy \(w_{1}\), and therefore, we find the maximization of \(H_{2}\) with respect to \(w_{2}\) and the minimization of \(H_{2}\) with respect to \(w_{1}.\) Consider the harvest function \(h(w_{1},w_{2})=w_{1}^{\tau }w_{2}^{\delta }\) with \(0\,<\,\tau\,<\,1\,<\,\delta. \)

Proposition 2

The optimal strategies of the 23 are given by
$$\begin{aligned} \left. \begin{array}{c} w_{1}=\left( \frac{\mu _{2} }{\delta \lambda _{2}}\right) ^{\frac{-\delta }{\delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}}\right) ^{\frac{\delta -1}{ (\delta -\tau -1)}} \\ w_{2}=\left( \frac{\mu _{2} }{\delta \lambda _{2}}\right) ^{\frac{\tau +1}{\delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}}\right) ^{^{\frac{-\tau }{ \delta -\tau -1}}} \\ h(w_{1},w_{2})=\left( \frac{\mu _{2} }{\delta \lambda _{2}}\right) ^{\frac{\delta }{ \delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}}\right) ^{^{\frac{ -\tau }{\delta -\tau -1}}} \end{array} \right\} \end{aligned}$$
(28)

Proof

From the necessary conditions  \(\frac{\partial H_{2}}{\partial w_{1}}=0,\)  \(\frac{\partial H_{2}}{\partial w_{2}}=0,\) we have
$$ h_{w_{1}}= \tau w_{1}^{\tau -1}w_{2}^{\delta }=\frac{\rho }{(w_{1}^2\lambda _{2})},\hbox {then} \,\,w_{1}=\left( \frac{\rho }{\tau \lambda _{2}}\right) ^\frac{1}{\tau +1}w_{2}^\frac{-\eta }{\tau +1} $$
(29)
and 
$$ h_{w_{2}}= \ \delta w_{1}^{\tau }v^{\delta -1}=\frac{\mu _{2}}{\lambda _{2}}, \hbox {then} \,\,w_{2}=\left( \frac{\mu _{2}}{\delta \lambda _{2}}\right) ^ \frac{1}{\delta -1}w_{1}^\frac{-\tau }{\delta -1} , $$
(30)
and thus
$$\begin{aligned} \left. \begin{array}{c} w_{1}=\left( \frac{\mu _{2} }{\delta \lambda _{2}}\right) ^{-\frac{\delta }{\delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}}\right) ^{\frac{\delta -1}{ (\delta -\tau -1)}} \\ w_{2} =\left( \frac{\mu _{2} }{\delta \lambda _{2}}\right) ^{\frac{\tau +1}{\delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}}\right) ^{^{\frac{-\tau }{ \delta -\tau -1}}} \\ h(w_{1},w_{2})=\left( \frac{\mu _{2} }{\delta \lambda _{2}}\right) ^{\frac{\delta }{ \delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}}\right) ^{^{\frac{ -\tau }{\delta -\tau -1}}} \end{array} \right\} \end{aligned}$$
(31)
\(\square \)
By using Proposition 2, we have
$$\begin{aligned} \left. \begin{array}{c} \left| \begin{array}{cc} H_{2w_{1}w_{1}} & H_{2w_{1}w_{2}} \\ \\ H_{2w_{2}w_{1}} & H_{2w_{2}w_{2}} \end{array} \right| =\left| \begin{array}{cc} \tau (\tau -1)w_{1}^{\tau -2}w_{2}^{\delta } & \tau \delta w_{1}^{\tau -1}w_{2}^{\delta -1} \\ \\ \tau \delta w_{1}^{\tau -1}w_{2}^{\delta -1} & \delta (\delta -1)w_{1}^{\tau }w_{2}^{\delta -2} \end{array} \right| \\ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\tau \delta (1-\tau -\delta )w_{1}^{2(\tau -1)}w_{2}^{2(\delta -1)}<0 \end{array} \right\} \end{aligned}$$
(32)
and \((w_{1},w_{2})\) is saddle point of the 23. Similarly from the above Lemma 1, we have
$$\begin{aligned} \left. \begin{array}{l} y(t)=(y_{0}-\frac{1}{r}\left( \frac{\mu _{2} }{\delta \lambda _{2}}\right) ^{ \frac{\delta }{\delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}} \right) ^{^{\frac{-\tau }{\delta -\tau -1}}})e^{rt}+\frac{1}{r}\left( \frac{ \mu _{2} }{\delta \lambda _{2}}\right) ^{\frac{\delta }{\delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}}\right) ^{^{\frac{-\tau }{\delta -\tau -1}}} \\ J_{2}=\frac{\mu _{1} }{\eta _{2}-r}(y_{0}-\frac{1}{r}h(w_{1},w_{2}))+\frac{h(w_{1},w_{2})}{ r\eta _{2}}+\frac{\mu _{2} {w_{2}}}{\eta _{2}}-\frac{\rho }{\eta _{2}w_{1}} \end{array} \right\} \end{aligned}$$
(33)

4 Comparison

Now, we are going to make a comparison between the game of view’s of the government and the game of view’s of the ITO (Table 1).
Table 1

Game of view’s of the government and the game of view’s of the ITO

 

The game of view’s of the government

The game of view’s of the ITO

1

\({\mathrm {w}}_{1}=\left[ \left( \frac{\alpha }{\tau (c_{1}-\lambda _{1})}\right) ^{\delta-1}\left( \frac{c_{3}}{\delta (c_{1}-\lambda _{1})}\right) ^{-\delta}\right] ^{\frac{1}{1-\tau-\delta }}\)

\( {\mathrm {w}}_{1}=\left( \frac{\mu _{2} }{\delta \lambda _{2}}\right) ^{-\frac{\delta }{\delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}}\right) ^{\frac{ \delta -1}{(\delta -\tau -1)}}\)

2

\(w_{2}=\left[ \left( \frac{c_{3}}{\delta (c_{1}-\lambda _{1})}\right) ^{\tau-1}\left( \frac{\alpha }{\tau (c_{1}-\lambda _{1})}\right) ^{-\tau}\right] ^{\frac{1}{1-\tau-\delta }}\)

\(w_{2}=\left( \frac{ \mu _{2} }{\delta \lambda _{2}}\right) ^{\frac{\tau +1}{\delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}}\right) ^{^{\frac{-\tau }{\delta -\tau -1}}}\)

3

\( h(w_{1},w_{2})=\left[ \left( \frac{\alpha }{\tau (c_{1}-\lambda _{1})}\right) ^{-\tau}\left( \frac{c_{3}}{\delta (c_{1}-\lambda _{1})}\right) ^{-\delta}\right] ^{\frac{1}{1-\tau-\delta }}\)

\(h(w_{1},w_{2})=\left( \frac{\mu _{2} }{\delta \lambda _{2}}\right) ^{\frac{\delta }{\delta -\tau -1}}\left( \frac{\rho }{\tau \lambda _{2}} \right) ^{^{\frac{-\tau }{\delta -\tau -1}}}\)

4

\(y(t)=(y_{0}-\frac{1}{r}h(w_{1},w_{2}))e^{rt}+\frac{h}{r}\)

\(y(t)=(y_{0}-\frac{1 }{r}h(w_{1},w_{2}))e^{rt}+\frac{1}{r}h\)

5

\(J_{1}=\frac{h}{\eta _{1}}(c_{1}- \frac{c_{2}}{r-\eta _{1}})-\frac{c_{3}w_{2}}{\eta _{1}}-\frac{\alpha w_{1} }{\eta _{1}}+\frac{c_{2}y_{0}}{r-\eta _{1}}\)

\(J_{2}= \frac{\mu _{1} }{\eta _{2}-r(y}_{0}-\frac{1}{r}h(u,v))+\frac{h(u,v)}{r\eta _{2}}+\frac{\mu _{2} w_{2}}{\eta _{2}}-\frac{\rho }{\eta _{2}w_{1}}\)

This comparison shows to the optimal strategy, harvest function, state trajectory and the objective in the event that the government took the initiative and start attacking terrorist organizations, that is clear in the column of the game of view’s of the government, and vise versa if the terrorist organizations took the initiative and start attacking the government, also is shown in the column of the game of view’s of the ITO

5 Conclusion

In this work, we discussed a very important problem “Terrorism Problem” with considered it as the game between two players, the government as the player 1 and the ITO as the plater 2, we discussed this game of view’s the government and ITO and derived the saddle points, the objectives and the state variables for each game. Finally, we made a comparison between the game of view’s of the government and the game of view’s of the ITO. I hope from this study help the government in the fight against terrorist organizations.

Notes

Acknowledgements

My highly grateful and appreciation to the Basic Sciences Research Unit, Deanship of Scientific Research at Majmaah University for funding this study, Project No.23,1436 h-2015 ad.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no competing interest.

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Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  1. 1.Basic Science Department, Faculty of Computers and InformaticsSuez Canal UniversityIsmailiaEgypt
  2. 2.Mathematics Department, College of ScienceMajmaah UniversityZulfiSaudi Arabia

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