Mathematical Game Theory and Applications

Preface xi

Introduction xiii

1 Strategic-Form Two-Player Games 1

Introduction 1

1.1 The Cournot Duopoly 2

1.2 Continuous Improvement Procedure 3

1.3 The Bertrand Duopoly 4

1.4 The Hotelling Duopoly 5

1.5 The Hotelling Duopoly in 2D Space 6

1.6 The Stackelberg Duopoly 8

1.7 Convex Games 9

1.8 Some Examples of Bimatrix Games 12

1.9 Randomization 13

1.10 Games 2 ×2 16

1.11 Games 2 × n and m ×2 18

1.12 The Hotelling Duopoly in 2D Space with Non-Uniform Distribution of Buyers 20

1.13 Location Problem in 2D Space 25

Exercises 26

2 Zero-Sum Games 28

Introduction 28

2.1 Minimax and Maximin 29

2.2 Randomization 31

2.3 Games with Discontinuous Payoff Functions 34

2.4 Convex-Concave and Linear-Convex Games 37

2.5 Convex Games 39

2.6 Arbitration Procedures 42

2.7 Two-Point Discrete Arbitration Procedures 48

2.8 Three-Point Discrete Arbitration Procedures with Interval Constraint 53

2.9 General Discrete Arbitration Procedures 56

Exercises 62

3 Non-Cooperative Strategic-Form n-Player Games 64

Introduction 64

3.1 Convex Games. The Cournot Oligopoly 65

3.2 Polymatrix Games 66

3.3 Potential Games 69

3.4 Congestion Games 73

3.5 Player-Specific Congestion Games 75

3.6 Auctions 78

3.7 Wars of Attrition 82

3.8 Duels, Truels, and Other Shooting Accuracy Contests 85

3.9 Prediction Games 88

Exercises 93

4 Extensive-Form n-Player Games 96

Introduction 96

4.1 Equilibrium in Games with Complete Information 97

4.2 Indifferent Equilibrium 99

4.3 Games with Incomplete Information 101

4.4 Total Memory Games 105

Exercises 108

5 Parlor Games and Sport Games 111

Introduction 111

5.1 Poker. A Game-Theoretic Model 112

5.2 The Poker Model with Variable Bets 118

5.3 Preference. A Game-Theoretic Model 129

5.4 The Preference Model with Cards Play 136

5.5 Twenty-One. A Game-Theoretic Model 145

5.6 Soccer. A Game-Theoretic Model of Resource Allocation 147

Exercises 152

6 Negotiation Models 155

Introduction 155

6.1 Models of Resource Allocation 155

6.2 Negotiations of Time and Place of a Meeting 166

6.3 Stochastic Design in the Cake Cutting Problem 171

6.4 Models of Tournaments 182

6.5 Bargaining Models with Incomplete Information 190

6.6 Reputation in Negotiations 221

Exercises 228

7 Optimal Stopping Games 230

Introduction 230

7.1 Optimal Stopping Game: The Case of Two Observations 231

7.2 Optimal Stopping Game: The Case of Independent Observations 234

7.3 The Game ?N(G) Under N ? 3 237

7.4 Optimal Stopping Game with Random Walks 241

7.5 Best Choice Games 250

7.6 Best Choice Game with Stopping Before Opponent 254

7.7 Best Choice Game with Rank Criterion. Lottery 259

7.8 Best Choice Game with Rank Criterion. Voting 264

7.9 Best Mutual Choice Game 269

Exercises 276

8 Cooperative Games 278

Introduction 278

8.1 Equivalence of Cooperative Games 278

8.2 Imputations and Core 281

8.3 Balanced Games 285

8.4 The -Value of a Cooperative Game 286

8.5 Nucleolus 289

8.6 The Bankruptcy Game 293

8.7 The Shapley Vector 298

8.8 Voting Games. The Shapley-Shubik Power Index and the Banzhaf Power Index 302

8.9 The Mutual Influence of Players. The Hoede-Bakker Index 309

Exercises 312

9 Network Games 314

Introduction 314

9.1 The KP-Model of Optimal Routing with Indivisible Traffic. The Price of Anarchy 315

9.2 Pure Strategy Equilibrium. Braess's Paradox 316

9.3 Completely Mixed Equilibrium in the Optimal Routing Problem with Inhomogeneous Users and Homogeneous Channels 319

9.4 Completely Mixed Equilibrium in the Optimal Routing Problem with Homogeneous Users and Inhomogeneous Channels 320

9.5 Completely Mixed Equilibrium: The General Case 322

9.6 The Price of Anarchy in the Model with Parallel Channels and Indivisible Traffic 324

9.7 The Price of Anarchy in the Optimal Routing Model with Linear Social Costs and Indivisible Traffic for an Arbitrary Network 328

9.8 The Mixed Price of Anarchy in the Optimal Routing Model with Linear Social Costs and Indivisible Traffic for an Arbitrary Network 332

9.9 The Price of Anarchy in the Optimal Routing Model with Maximal Social Costs and Indivisible Traffic for an Arbitrary Network 335

9.10 The Wardrop Optimal Routing Model with Divisible Traffic 337

9.11 The Optimal Routing Model with Parallel Channels. The Pigou Model. Braess's Paradox 340

9.12 Potential in the Optimal Routing Model with Indivisible Traffic for an Arbitrary Network 341

9.13 Social Costs in the Optimal Routing Model with Divisible Traffic for Convex Latency Functions 343

9.14 The Price of Anarchy in the Optimal Routing Model with Divisible Traffic for Linear Latency Functions 344

9.15 Potential in the Wardrop Model with Parallel Channels for Player-Specific Linear Latency Functions 346

9.16 The Price of Anarchy in an Arbitrary Network for Player-Specific Linear Latency Functions 349

Exercises 351

10 Dynamic Games 352

Introduction 352

10.1 Discrete-Time Dynamic Games 353

10.2 Some Solution Methods for Optimal Control Problems with One Player 358

10.3 The Maximum Principle and the Bellman Equation in Discrete- and Continuous-Time Games of N Players 368

10.4 The Linear-Quadratic Problem on Finite and Infinite Horizons 375

10.5 Dynamic Games in Bioresource Management Problems. The Case of Finite Horizon 378

10.6 Dynamic Games in Bioresource Management Problems. The Case of Infinite Horizon 383

10.7 Time-Consistent Imputation Distribution Procedure 388

Exercises 402

References 405

Index 411