Acknowledgments ix

Contents

Acknowledgments ix

Abstract xv

R´ esum´ e xvii

Resumen xix

1 Introduction 1

1.1 Game Theory . . . . 4

1.2 Bilevel Programming . . . . 10

2 Problem definition 13 2.1 General Stackelberg games–GSGs . . . . 13

2.2 Stackelberg security games–SSGs . . . . 14

2.2.1 Recovering an optimal mixed strategy for the defender–The box method 17 3 State of the art 21 3.1 Computational complexity . . . . 21

3.2 Computational challenges . . . . 23

3.3 Methods . . . . 24

3.4 Noteworthy extensions . . . . 28

4 A study of general and security Stackelberg game formulations 31 4.1 Introduction . . . . 31

4.2 General Stackelberg games–GSGs . . . . 32

4.2.1 Single level MILP formulations . . . . 32

4.2.2 Comparison of the formulations . . . . 35

4.2.3 Computational experiments for GSGs . . . . 38

4.3 Stackelberg security games–SSGs . . . . 40

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4.3.1 Single level MILP formulations . . . . 40

4.3.2 Comparison of the formulations . . . . 47

4.3.3 Computational experiments for SSGs . . . . 50

4.4 Conclusions . . . . 54

5 Benders decomposition methods 55 5.1 Introduction . . . . 55

5.2 Benders decomposition . . . . 57

5.3 Decomposition approaches . . . . 59

5.3.1 General Stackelberg games . . . . 60

5.3.2 Stackelberg security games . . . . 62

5.4 Cutting plane approach . . . . 64

5.5 Implementation considerations . . . . 66

5.5.1 Root node cut loop stabilization . . . . 66

5.5.2 Primal lower bound-enhancing heuristics . . . . 68

5.5.3 Primal upper bound-enhancing heuristics . . . . 72

5.6 Computational experiments . . . . 73

5.6.1 Configuring the cutting plane approaches . . . . 74

5.6.2 Comparing the performance of the cutting plane approaches . . . . . 89

5.7 Conclusions . . . . 96

6 Stackelberg security games with combined defender strategies 99 6.1 Introduction . . . . 99

6.2 Problem definition . . . . 100

6.3 Solving the problem . . . . 101

6.3.1 The formulation: (FENCE

) . . . . 102

6.3.2 Recovering an implementable defender mixed strategy . . . . 105

6.4 Computational experiments . . . . 109

6.4.1 Performance of the two-stage sampling method . . . . 109

6.4.2 Performance of (FENCE

) . . . . 110

6.5 Conclusions . . . . 111

7 Case Study: Border Protection 113 7.1 Introduction . . . . 113

7.2 The border patrol problem . . . . 115

7.3 Parameter generation: Constructing the game . . . . 116

7.4 Building software for Carabineros . . . . 119

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7.4.1 Parameter estimation software . . . . 119

7.4.2 Deployment generation software . . . . 120

7.5 Computational tests . . . . 121

7.6 Evaluating our deployed system . . . . 122

7.6.1 Evaluating the mathematical model . . . . 122

7.6.2 Evaluating the software . . . . 124

7.7 Conclusions . . . . 124

8 Conclusions and future work 125 8.1 Summary of main results . . . . 125

8.2 Perspectives . . . . 127

8.3 Closing remarks . . . . 128

Bibliography 129 Appendix A Additional material 141 A.1 (QUAD

) is equivalent to (MIP-p-G

) . . . . 141

A.2 Tight M values . . . . 143

A.2.1 General Stackelberg games . . . . 143

A.2.2 Stackelberg security games . . . . 145

A.3 (MIP-p-G) is convex hull defining for p = 1 . . . . 147

A.4 E↵ect of primal lower bound heuristics on GSG cutting plane approaches . 149 A.5 E↵ect of primal lower bound heuristics on SSG cutting plane approaches . . 150

A.6 Recovery of an implementable defender strategy x

from (FENCE

) 151 A.7 Determining action areas along the border using QGIS . . . . 153

List of Tables 159

List of Figures 163

Vita 165

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