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

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## 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

*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

**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

**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

*r*, i.e.,

## 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

### **Definition 1**

### 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*

*such that the following conditions are satisfied*

### *Proof*

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

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*

*with the harvest function*

### *Proof*

### **Lemma 1**

*The objective of the government (player 1) for the constant strategies*\(w_{1},w_{2}\)

*is*

### *Proof*

### 3.3 The game of view’s of ITO

### **Definition 2**

### **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*

*such that the following conditions are satisfied*

### *Proof*

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

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*

### *Proof*

## 4 Comparison

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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