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Preamble xi I INTRODUCTION 1 Noncooperative Games 1.1 Elements of a Game 3 1.2 Cooperative vs. Noncooperative Games: Rope-Pulling 4 1.3 Robust Designs: Resistive Circuit 8 1.4 Mixed Policies: Network Routing 9 1.5 Nash Equilibrium 11 1.6 Practice Exercise 11 2 Policies 2.1 Actions vs. Policies: Advertising Campaign 13 2.2 Multi-Stage Games:War of Attrition 16 2.3 Open vs. Closed-Loop: Zebra in the Lake 18 2.4 Practice Exercises 19 II ZERO-SUM GAMES 3 Zero-Sum Matrix Games 3.1 Zero-Sum Matrix Games 25 3.2 Security Levels and Policies 26 3.3 Computing Security Levels and Policies with MATLAB(R) 27 3.4 Security vs. Regret: Alternate Play 28 3.5 Security vs. Regret: Simultaneous Plays 28 3.6 Saddle-Point Equilibrium 29 3.7 Saddle-Point Equilibrium vs. Security Levels 30 3.8 Order Interchangeability 32 3.9 Computational Complexity 32 3.10 Practice Exercise 34 3.11 Additional Exercise 34 4 Mixed Policies 4.1 Mixed Policies: Rock-Paper-Scissor 35 4.2 Mixed Action Spaces 37 4.3 Mixed Security Policies and Saddle-Point Equilibrium 38 4.4 Mixed Saddle-Point Equilibrium vs. Average Security Levels 41 4.5 General Zero-Sum Games 43 4.6 Practice Exercises 47 4.7 Additional Exercise 50 5 Minimax Theorem 5.1 Theorem Statement 52 5.2 Convex Hull 53 5.3 Separating Hyperplane Theorem 54 5.4 On theWay to Prove the Minimax Theorem 55 5.5 Proof of the Minimax Theorem 57 5.6 Consequences of the Minimax Theorem 58 5.7 Practice Exercise 58 6 Computation of Mixed Saddle-Point Equilibrium Policies 6.1 Graphical Method 60 6.2 Linear Program Solution 61 6.3 Linear Programs with MATLAB(R) 63 6.4 Strictly Dominating Policies 64 6.5 "Weakly" Dominating Policies 66 6.6 Practice Exercises 67 6.7 Additional Exercise 70 7 Games in Extensive Form 7.1 Motivation 71 7.2 Extensive Form Representation 72 7.3 Multi-Stage Games 72 7.4 Pure Policies and Saddle-Point Equilibria 74 7.5 Matrix Form for Games in Extensive Form 75 7.6 Recursive Computation of Equilibria for Single-Stage Games 77 7.7 Feedback Games 79 7.8 Feedback Saddle-Point for Multi-Stage Games 79 7.9 Recursive Computation of Equilibria for Multi-Stage Games 83 7.10 Practice Exercise 85 7.11 Additional Exercises 86 8 Stochastic Policies for Games in Extensive Form 8.1 Mixed Policies and Saddle-Point Equilibria 87 8.2 Behavioral Policies for Games in Extensive Form 90 8.3 Behavioral Saddle-Point Equilibria 91 8.4 Behavioral vs. Mixed Policies 92 8.5 Recursive Computation of Equilibria for Feedback Games 93 8.6 Mixed vs. Behavioral Order Interchangeability 95 8.7 Non-Feedback Games 95 8.8 Practice Exercises 96 8.9 Additional Exercises 102 III NON-ZERO-SUM GAMES 9 Two-Player Non-Zero-Sum Games 9.1 Security Policies and Nash Equilibria 105 9.2 Bimatrix Games 107 9.3 Admissible Nash Equilibria 108 9.4 Mixed Policies 110 9.5 Best-Response Equivalent Games and Order Interchangeability 111 9.6 Practice Exercises 114 9.7 Additional Exercises 116 10 Computation of Nash Equilibria for Bimatrix Games 10.1 Completely Mixed Nash Equilibria 118 10.2 Computation of Completely Mixed Nash Equilibria 120 10.3 Numerical Computation of Mixed Nash Equilibria 121 10.4 Practice Exercise 124 10.5 Additional Exercise 126 11 N-Player Games 11.1 N-Player Games 127 11.2 Pure N-Player Games in Normal Form 129 11.3 Mixed Policies for N-Player Games in Normal Form 130 11.4 Completely Mixed Policies 131 12 Potential Games 12.1 Identical Interests Games 133 12.2 Potential Games 135 12.3 Characterization of Potential Games 138 12.4 Potential Games with Interval Action Spaces 139 12.5 Practice Exercises 142 12.6 Additional Exercise 144 13 Classes of Potential Games 13.1 Identical Interests Plus Dummy Games 145 13.2 Decoupled Plus Dummy Games 146 13.3 Bilateral Symmetric Games 147 13.4 Congestion Games 148 13.5 Other Potential Games 149 13.6 Distributed Resource Allocation 150 13.7 Computation of Nash Equilibria for Potential Games 153 13.8 Fictitious Play 156 13.9 Practice Exercises 159 13.10 Additional Exercises 167 IV DYNAMIC GAMES 14 Dynamic Games 14.1 Game Dynamics 171 14.2 Information Structures 173 14.3 Continuous-Time Differential Games 175 14.4 Differential Games with Variable Termination Time 177 15 One-Player Dynamic Games 15.1 One-Player Discrete-Time Games 178 15.2 Discrete-Time Cost-To-Go 179 15.3 Discrete-Time Dynamic Programming 179 15.4 Computational Complexity 184 15.5 Solving Finite One-Player Games with MATLAB(R) 186 15.6 Linear Quadratic Dynamic Games 187 15.7 Practice Exercise 187 15.8 Additional Exercise 189 16 One-Player Differential Games 16.1 One-Player Continuous-Time Differential Games 190 16.2 Continuous-Time Cost-To-Go 191 16.3 Continuous-Time Dynamic Programming 191 16.4 Linear Quadratic Dynamic Games 195 16.5 Differential Games with Variable Termination Time 196 16.6 Practice Exercise 198 17 State-Feedback Zero-Sum Dynamic Games 17.1 Zero-Sum Dynamic Games in Discrete Time 201 17.2 Discrete-Time Dynamic Programming 203 17.3 Solving Finite Zero-Sum Games with MATLAB(R) 205 17.4 Linear Quadratic Dynamic Games 206 17.5 Practice Exercise 209 18 State-Feedback Zero-Sum Differential Games 18.1 Zero-Sum Dynamic Games in Continuous Time 214 18.2 Linear Quadratic Dynamic Games 216 18.3 Differential Games with Variable Termination Time 219 18.4 Pursuit-Evasion 220 18.5 Practice Exercise 222 References 223 Index 225