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Dynamic Noncooperative Game Theory (Classics in Applied Mathematics) (Paperback)
by Tamer Basar (Author), Geert Jan Olsder (Author)
Product Description
Recent interest in biological games and mathematical finance make this classic 1982 text a necessity once again. Unlike other books in the field, this text provides an overview of the analysis of dynamic/differential zero-sum and nonzero-sum games and simultaneously stresses the role of different information patterns. The first edition was fully revised in 1995, adding new topics such as randomized strategies, finite games with integrated decisions, and refinements of Nash equilibrium. Readers can now look forward to even more recent results in this unabridged, revised SIAM Classics edition. Topics covered include static and dynamic noncooperative game theory, with an emphasis on the interplay between dynamic information patterns and structural properties of several different types of equilibria; Nash and Stackelberg solution concepts; multi-act games; Braess paradox; differential games; the relationship between the existence of solutions of Riccati equations and the existence of Nash equilibrium solutions; and infinite-horizon differential games.
Book Description
This text provides an overview of the analysis of dynamic/differential zero-sum and nonzero-sum games and simultaneously stresses the role of different information patterns. Fully revised in 1995, this edition features new topics such as randomized strategies, finite games with integrated decisions, and refinements of Nash equilibrium.
Recent interest in biological games and mathematical finance make this classic 1982 text a necessity once again. Unlike other books in the field, this text provides an overview of the analysis of dynamic/differential zero-sum and nonzero-sum games and simultaneously stresses the role of different information patterns. The first edition was fully revised in 1995, adding new topics such as randomized strategies, finite games with integrated decisions, and refinements of Nash equilibrium. Readers can now look forward to even more recent results in this unabridged, revised SIAM Classics edition. Topics covered include static and dynamic noncooperative game theory, with an emphasis on the interplay between dynamic information patterns and structural properties of several different types of equilibria; Nash and Stackelberg solution concepts; multi-act games; Braess paradox; differential games; the relationship between the existence of solutions of Riccati equations and the existence of Nash equilibrium solutions; and infinite-horizon differential games.
Book Description
This text provides an overview of the analysis of dynamic/differential zero-sum and nonzero-sum games and simultaneously stresses the role of different information patterns. Fully revised in 1995, this edition features new topics such as randomized strategies, finite games with integrated decisions, and refinements of Nash equilibrium.
Contents
Preface to the Classics Edition xi
Preface to the Second Edition xiii
1 Introduction and Motivation 1
1.1 Preliminary Remarks 1
1.2 Preview on Noncooperative Games 3
1.3 Outline of the Book 12
1.4 Conventions, Notation and Terminology 13
Part I
2 Noncooperative Finite Games: Two-Person Zero-Sum 17
2.1 Introduction 17
2.2 Matrix Games 18
2.3 Computation of Mixed Equilibrium Strategies 29
2.4 Extensive Forms: Single-Act Games 36
2.5 Extensive Games: Multi-Act Games 45
2.6 Zero-Sum Games with Chance Moves 57
2.7 Two Extensions 60
2.7.1 Games with repeated decisions 61
2.7.2 Extensive forms with cycles 63
2.8 Action-Dependent Information Sets 65
2.8.1 Duels 66
2.8.2 A searchlight game 66
2.9 Problems 70
2.10 Notes 75
3 Noncooperative Finite Games: N-Person Nonzero-Sum 77
3.1 Introduction 77
3.2 Bimatrix Games 78
3.3 N-Person Games in Normal Form 88
3.4 Computation of Mixed-Strategy Nash Equilibria in Bimatrix Games 95
3.5 Nash Equilibria of N-Person Games in Extensive Form 97
3.5.1 Single-act games: Pure-strategy Nash equilibria 100
3.5.2 Single-act games: Nash equilibria in behavioral and mixed
strategies 116
3.5.3 Multi-act games: Pure-strategy Nash equilibria 118
3.5.4 Multi-act games: Behavioral and mixed equilibrium
strategies 126
3.5.5 Other refinements on Nash equilibria 128
3.6 The Stackelberg Equilibrium Solution 131
3.7 Nonzero-Sum Games with Chance Moves 148
3.8 Problems 153
3.9 Notes 159
4 Static Noncooperative Infinite Games 161
4.1 Introduction 161
4.2 e Equilibrium Solutions 162
4.3 Continuous-Kernel Games: Reaction Curves, and Existence and
Uniqueness of Nash and Saddle-Point Equilibria 168
4.4 Stackelberg Solution of Continuous-Kernel Games 179
4.5 Consistent Conjectural Variations Equilibrium 186
4.6 Quadratic Games with Applications in Microeconomics 190
4.7 Braess Paradox 203
4.8 Problems 205
4.9 Notes 210
Part II
5 General Formulation of Infinite Dynamic Games 215
5.1 Introduction 215
5.2 Discrete-Time Infinite Dynamic Games 216
5.3 Continuous-Time Infinite Dynamic Games 224
5.4 Mixed and Behavioral Strategies in Infinite Dynamic Games . . 230
5.5 Tools for One-Person Optimization 233
5.5.1 Dynamic programming for discrete-time systems 233
5.5.2 Dynamic programming for continuous-time systems . . . 236
5.5.3 The minimum principle 241
5.6 Representations of Strategies Along Trajectories, and Time Consistency
of Optimal Policies 247
5.7 Viscosity Solutions 255
5.8 Problems 260
5.9 Notes 262
6 Nash and Saddle-Point Equilibria of Infinite Dynamic Games 265
6.1 Introduction 265
6.2 Open-Loop and Feedback Nash and Saddle-Point Equilibria for
Dynamic Games in Discrete Time 266
CONTENTS ix
6.2.1 Open-loop Nash equilibria 267
6.2.2 Closed-loop no-memory and feedback Nash equilibria . . . 276
6.2.3 Linear-quadratic games with an infinite number of stages 288
6.3 Informational Properties of Nash Equilibria in Discrete-Time
Dynamic Games 292
6.3.1 A three-person dynamic game illustrating informational
nonuniqueness 292
6.3.2 General results on informationally nonunique equilibrium
solutions 296
6.4 Stochastic Nonzero-Sum Games with Deterministic Information
Patterns 303
6.5 Open-Loop and Feedback Nash and Saddle-Point Equilibria of
Differential Games 310
6.5.1 Open-loop Nash equilibria 310
6.5.2 Closed-loop no-memory and feedback Nash equilibria . . . 320
6.5.3 Linear-quadratic differential games on an infinite time
horizon 333
6.6 Applications in Robust Controller Designs: H°°-Optimal Control 342
6.7 Stochastic Differential Games with Deterministic Information Patterns
350
6.8 Problems 355
6.9 Notes 361
7 Stackelberg Equilibria of Infinite Dynamic Games 365
7.1 Introduction 365
7.2 Open-Loop Stackelberg Solution of Two-Person Dynamic Games
in Discrete Time 366
7.3 Feedback Stackelberg Solution Under CLPS Information Pattern 373
7.4 (Global) Stackelberg Solution Under CLPS Information Pattern 376
7.4.1 An illustrative example (Example 7.1) 376
7.4.2 A second example (Example 7.2): Follower acts twice in
the game 382
7.4.3 Linear Stackelberg solution of linear-quadratic dynamic
games 385
7.4.4 Incentives (deterministic) 392
7.5 Stochastic Dynamic Games with Deterministic Information Patterns 396
7.5.1 (Global) Stackelberg solution 396
7.5.2 Feedback Stackelberg solution 402
7.5.3 Stochastic incentive problems 403
7.6 Stackelberg Solution of Differential Games 407
7.6.1 The open-loop information structure 407
7.6.2 The CLPS information pattern 412
7.7 Problems 418
7.8 Notes 421
8 Pursuit-Evasion Games 423
8.1 Introduction 423
8.2 Necessary and Sufficient Conditions for Saddle-Point Equilibria . 424
8.2.1 The Isaacs equation 425
8.2.2 Upper and lower values, and viscosity solutions 432
8.3 Capturability 434
8.4 Singular Surfaces 442
8.5 Solution of a Pursuit-Evasion Game: The Lady in the Lake . . . 448
8.6 An Application in Maritime Collision Avoidance 451
8.7 Role Determination and an Application in Aeronautics 456
8.8 Problems 464
8.9 Notes 467
Appendix A Mathematical Review 471
A.1 Sets 471
A.2 Normed Linear (Vector) Spaces 472
A.3 Matrices 473
A.4 Convex Sets andFunctionals 473
A.5 Optimization of Functionals 474
Appendix B Some Notions of Probability Theory 477
B.1 Ingredients of Probability Theory 477
B.2 Random Vectors 478
B.3 Integrals and Expectation 480
B.4 Norms and the Cauchy-Schwarz Inequality 481
Appendix C Fixed Point Theorems 483
Bibliography 485
List of Corollaries, Definitions, Examples, Lemmas, Propositions,
Remarks and Theorems 507
Index 515
Preface to the Classics Edition xi
Preface to the Second Edition xiii
1 Introduction and Motivation 1
1.1 Preliminary Remarks 1
1.2 Preview on Noncooperative Games 3
1.3 Outline of the Book 12
1.4 Conventions, Notation and Terminology 13
Part I
2 Noncooperative Finite Games: Two-Person Zero-Sum 17
2.1 Introduction 17
2.2 Matrix Games 18
2.3 Computation of Mixed Equilibrium Strategies 29
2.4 Extensive Forms: Single-Act Games 36
2.5 Extensive Games: Multi-Act Games 45
2.6 Zero-Sum Games with Chance Moves 57
2.7 Two Extensions 60
2.7.1 Games with repeated decisions 61
2.7.2 Extensive forms with cycles 63
2.8 Action-Dependent Information Sets 65
2.8.1 Duels 66
2.8.2 A searchlight game 66
2.9 Problems 70
2.10 Notes 75
3 Noncooperative Finite Games: N-Person Nonzero-Sum 77
3.1 Introduction 77
3.2 Bimatrix Games 78
3.3 N-Person Games in Normal Form 88
3.4 Computation of Mixed-Strategy Nash Equilibria in Bimatrix Games 95
3.5 Nash Equilibria of N-Person Games in Extensive Form 97
3.5.1 Single-act games: Pure-strategy Nash equilibria 100
3.5.2 Single-act games: Nash equilibria in behavioral and mixed
strategies 116
3.5.3 Multi-act games: Pure-strategy Nash equilibria 118
3.5.4 Multi-act games: Behavioral and mixed equilibrium
strategies 126
3.5.5 Other refinements on Nash equilibria 128
3.6 The Stackelberg Equilibrium Solution 131
3.7 Nonzero-Sum Games with Chance Moves 148
3.8 Problems 153
3.9 Notes 159
4 Static Noncooperative Infinite Games 161
4.1 Introduction 161
4.2 e Equilibrium Solutions 162
4.3 Continuous-Kernel Games: Reaction Curves, and Existence and
Uniqueness of Nash and Saddle-Point Equilibria 168
4.4 Stackelberg Solution of Continuous-Kernel Games 179
4.5 Consistent Conjectural Variations Equilibrium 186
4.6 Quadratic Games with Applications in Microeconomics 190
4.7 Braess Paradox 203
4.8 Problems 205
4.9 Notes 210
Part II
5 General Formulation of Infinite Dynamic Games 215
5.1 Introduction 215
5.2 Discrete-Time Infinite Dynamic Games 216
5.3 Continuous-Time Infinite Dynamic Games 224
5.4 Mixed and Behavioral Strategies in Infinite Dynamic Games . . 230
5.5 Tools for One-Person Optimization 233
5.5.1 Dynamic programming for discrete-time systems 233
5.5.2 Dynamic programming for continuous-time systems . . . 236
5.5.3 The minimum principle 241
5.6 Representations of Strategies Along Trajectories, and Time Consistency
of Optimal Policies 247
5.7 Viscosity Solutions 255
5.8 Problems 260
5.9 Notes 262
6 Nash and Saddle-Point Equilibria of Infinite Dynamic Games 265
6.1 Introduction 265
6.2 Open-Loop and Feedback Nash and Saddle-Point Equilibria for
Dynamic Games in Discrete Time 266
CONTENTS ix
6.2.1 Open-loop Nash equilibria 267
6.2.2 Closed-loop no-memory and feedback Nash equilibria . . . 276
6.2.3 Linear-quadratic games with an infinite number of stages 288
6.3 Informational Properties of Nash Equilibria in Discrete-Time
Dynamic Games 292
6.3.1 A three-person dynamic game illustrating informational
nonuniqueness 292
6.3.2 General results on informationally nonunique equilibrium
solutions 296
6.4 Stochastic Nonzero-Sum Games with Deterministic Information
Patterns 303
6.5 Open-Loop and Feedback Nash and Saddle-Point Equilibria of
Differential Games 310
6.5.1 Open-loop Nash equilibria 310
6.5.2 Closed-loop no-memory and feedback Nash equilibria . . . 320
6.5.3 Linear-quadratic differential games on an infinite time
horizon 333
6.6 Applications in Robust Controller Designs: H°°-Optimal Control 342
6.7 Stochastic Differential Games with Deterministic Information Patterns
350
6.8 Problems 355
6.9 Notes 361
7 Stackelberg Equilibria of Infinite Dynamic Games 365
7.1 Introduction 365
7.2 Open-Loop Stackelberg Solution of Two-Person Dynamic Games
in Discrete Time 366
7.3 Feedback Stackelberg Solution Under CLPS Information Pattern 373
7.4 (Global) Stackelberg Solution Under CLPS Information Pattern 376
7.4.1 An illustrative example (Example 7.1) 376
7.4.2 A second example (Example 7.2): Follower acts twice in
the game 382
7.4.3 Linear Stackelberg solution of linear-quadratic dynamic
games 385
7.4.4 Incentives (deterministic) 392
7.5 Stochastic Dynamic Games with Deterministic Information Patterns 396
7.5.1 (Global) Stackelberg solution 396
7.5.2 Feedback Stackelberg solution 402
7.5.3 Stochastic incentive problems 403
7.6 Stackelberg Solution of Differential Games 407
7.6.1 The open-loop information structure 407
7.6.2 The CLPS information pattern 412
7.7 Problems 418
7.8 Notes 421
8 Pursuit-Evasion Games 423
8.1 Introduction 423
8.2 Necessary and Sufficient Conditions for Saddle-Point Equilibria . 424
8.2.1 The Isaacs equation 425
8.2.2 Upper and lower values, and viscosity solutions 432
8.3 Capturability 434
8.4 Singular Surfaces 442
8.5 Solution of a Pursuit-Evasion Game: The Lady in the Lake . . . 448
8.6 An Application in Maritime Collision Avoidance 451
8.7 Role Determination and an Application in Aeronautics 456
8.8 Problems 464
8.9 Notes 467
Appendix A Mathematical Review 471
A.1 Sets 471
A.2 Normed Linear (Vector) Spaces 472
A.3 Matrices 473
A.4 Convex Sets andFunctionals 473
A.5 Optimization of Functionals 474
Appendix B Some Notions of Probability Theory 477
B.1 Ingredients of Probability Theory 477
B.2 Random Vectors 478
B.3 Integrals and Expectation 480
B.4 Norms and the Cauchy-Schwarz Inequality 481
Appendix C Fixed Point Theorems 483
Bibliography 485
List of Corollaries, Definitions, Examples, Lemmas, Propositions,
Remarks and Theorems 507
Index 515
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