Game theory is the theory of social situations, and the majority of research into the topic focuses on how groups of people interact by developing formulas and algorithms to identify optimal strategies and to predict the outcome of interactions. Only fifty years old, it has already revolutionized economics and finance, and is spreading rapidly to a wide variety of fields. LQ Dynamic Optimization and Differential Games is an assessment of the state of the art in its field and the first modern book on linear-quadratic game theory, one of the most commonly used tools for modelling and analysing strategic decision making problems in economics and management. Linear quadratic dynamic models have a long tradition in economics, operations research and control engineering; and the author begins by describing the one-decision maker LQ dynamic optimization problem before introducing LQ differential games.
- Covers cooperative and non-cooperative scenarios, and treats the standard information structures (open-loop and feedback).
- Includes real-life economic examples to illustrate theoretical concepts and results.
- Presents problem formulations and sound mathematical problem analysis.
- Includes exercises and solutions, enabling use for self-study or as a course text.
- Supported by a website featuring solutions to exercises, further examples and computer code for numerical examples.
LQ Dynamic Optimization and Differential Games offers a comprehensive introduction to the theory and practice of this extensively used class of economic models, and will appeal to applied mathematicians and econometricians as well as researchers and senior undergraduate/graduate students in economics, mathematics, engineering and management science.
Table of Contents:
Preface ix
Notation and symbols xi
1 Introduction 1
1.1 Historical perspective 1
1.2 How to use this book 10
1.3 Outline of this book 10
1.4 Notes and references 14
2 Linear algebra 15
2.1 Basic concepts in linear algebra 15
2.2 Eigenvalues and eigenvectors 21
2.3 Complex eigenvalues 23
2.4 Cayley–Hamilton theorem 31
2.5 Invariant subspaces and Jordan canonical form 34
2.6 Semi-definite matrices 42
2.7 Algebraic Riccati equations 43
2.8 Notes and references 54
2.9 Exercises 55
2.10 Appendix 59
3 Dynamical systems 63
3.1 Description of linear dynamical systems 64
3.2 Existence–uniqueness results for differential equations 70
3.2.1 General case 70
3.2.2 Control theoretic extensions 74
3.3 Stability theory: general case 78
3.4 Stability theory of planar systems 83
3.5 Geometric concepts 91
3.6 Performance specifications 96
3.7 Examples of differential games 105
3.8 Information, commitment and strategies 114
3.9 Notes and references 114
3.10 Exercises 115
3.11 Appendix 118
4 Optimization techniques 121
4.1 Optimization of functions 121
4.2 The Euler–Lagrange equation 125
4.3 Pontryagin’s maximum principle 133
4.4 Dynamic programming principle 154
4.5 Solving optimal control problems 162
4.6 Notes and references 162
4.7 Exercises 163
4.8 Appendix 170
5 Regular linear quadratic optimal control 175
5.1 Problem statement 175
5.2 Finite-planning horizon 177
5.3 Riccati differential equations 192
5.4 Infinite-planning horizon 196
5.5 Convergence results 209
5.6 Notes and references 218
5.7 Exercises 219
5.8 Appendix 224
6 Cooperative games 229
6.1 Pareto solutions 230
6.2 Bargaining concepts 240
6.3 Nash bargaining solution 246
6.4 Numerical solution 251
6.5 Notes and references 253
6.6 Exercises 254
6.7 Appendix 259
7 Non-cooperative open-loop information games 261
7.1 Introduction 264
7.2 Finite-planning horizon 265
7.3 Open-loop Nash algebraic Riccati equations 278
7.4 Infinite-planning horizon 283
7.5 Computational aspects and illustrative examples 299
7.6 Convergence results 305
7.7 Scalar case 312
7.8 Economics examples 319
7.8.1 A simple government debt stabilization game 320
7.8.2 A game on dynamic duopolistic competition 322
7.9 Notes and references 326
7.10 Exercises 327
7.11 Appendix 340
8 Non-cooperative feedback information games 359
8.1 Introduction 359
8.2 Finite-planning horizon 362
8.3 Infinite-planning horizon 371
8.4 Two-player scalar case 383
8.5 Computational aspects 389
8.5.1 Preliminaries 390
8.5.2 A scalar numerical algorithm: the two-player case 393
8.5.3 The N-player scalar case 399
8.6 Convergence results for the two-player scalar case 403
8.7 Notes and references 412
8.8 Exercises 413
8.9 Appendix 421
9 Uncertain non-cooperative feedback information games 427
9.1 Stochastic approach 428
9.2 Deterministic approach: introduction 433
9.3 The one-player case 435
9.4 The one-player scalar case 444
9.5 The two-player case 450
9.6 A fishery management game 455
9.7 A scalar numerical algorithm 461
9.8 Stochastic interpretation 472
9.9 Notes and references 474
9.10 Exercises 475
9.11 Appendix 481
References 485
Index 495
