During the height of the Cold War, between 1965 and 1968, Robert Aumann, Michael Maschler, and Richard Stearns collaborated on research on the dynamics of arms control negotiations that has since become foundational to work on repeated games. These five seminal papers are collected here for the first time, with the addition of "postscripts" describing many of the developments since the papers were written. The basic model studied throughout the book is one in which players ignorant about the game being played must learn what they can from the actions of the others.The original work, done under contract to the United States Arms Control and Disarmament Agency, was intended to tackle the gradual disarmament problem, in which neither player knew what his own payoff would be for any given agreement, because of uncertainty about the other side's arsenal and weapons production technology. But the research soon became much more generalized,covering information concealment and revelation, signaling and learning, and related ideas in any repeated competitive situation.The first four chapters of the book treat the competitive zero-sum side of the theory of repeated games. Chapter five takes up cooperative phenomena where one player may want to signal information to another. An extensive bibliography covers all items mentioned in the main text, in the postscripts, and in the introduction. The bibliography also includes a compilation of published papers and books that refer to the original reports.
| Preface | ||
| Ch. 1 | Game Theoretic Aspects of Gradual Disarmament | |
| 2 | A Case When a Player Cannot Benefit in the Long Run from His Information | 3 |
| 3 | A Case When a Player Should Disclose His Information | 10 |
| 4 | A Case When a Player Should Partially Disclose His Information | 11 |
| 5 | The Main Theorem | 22 |
| 6 | The Optimal Strategy of Player 2 | 34 |
| 7 | Conclusions | |
| 8 | Where Do We Go From Here? | 39 |
| Postscript a Martingales | 42 | |
| Postscript b Convexity and Concavity as Monotonicity in Information | 43 | |
| Postscript c The Error Term | 44 | |
| Postscript d Optimal Strategies of the Uninformed Player | 53 | |
| Postscript e Monotonicity of v[subscript n] and the Recursive Structure of [actual symbol not reproducible] | 61 | |
| Ch. 2 | Repeated Games with Incomplete Information: A Survey of Recent Results | |
| 2 | Repeated Games and the Problem of Information | 66 |
| 3 | Lack of Information on One Side - Stage Games in Normal Form | 70 |
| 4 | Lack of Information on Both Sides | 91 |
| 5 | Incomplete Knowledge of Moves | 110 |
| 6 | Further Problems | |
| Postscript a Incomplete Information on Both Sides: lim v[subscript n] | 130 | |
| Postscript b Incomplete Information on Both Sides: The Limit of the Discounted Values | 130 | |
| Postscript c The Conceptual Distinction Between [actual symbol not reproducible]: Generalities | 131 | |
| Postscript d The Conceptual Distinction Between [actual symbol not reproducible]: Incomplete Information | 136 | |
| Postscript e Non-Zero Sum Games | 137 | |
| Postscript f Discounting | 139 | |
| Postscript g Continuous Time | 140 | |
| Postscript h Alternative Definitions of Value, Maxmin, and Minmax in [actual symbol not reproducible] | 140 | |
| Postscript i Incomplete Information on Both Sides: The Dependent Case | 143 | |
| Postscript j Incomplete Information on One-and-a-Half Sides | 145 | |
| Ch. 3 | A Formal Information Concept for Games with Incomplete Information | 155 |
| Ch. 4 | Repeated Games of Incomplete Information: The Zero-Sum Extensive Case | |
| 2 | Motivation | 176 |
| 3 | The Mathematical Model | 177 |
| 4 | Examples | 181 |
| 5 | The Infinite-Stage Game and its Relation to the n-Stage Games | 186 |
| 6 | Statement of the Main Theorem: The Value of the Infinite-Stage Game | 188 |
| 7 | Proof of the Main Theorem | 191 |
| Postscript a Monotonicity of v[subscript n] and the Informed Player's Information Matrices | 206 | |
| Postscript b Remembering One's Own Moves | 211 | |
| Postscript c Optimal Strategies for the Uninformed Player | 212 | |
| Postscript d The Case When P2 Knows His Payoff | 212 | |
| Postscript e The Error Term | 213 | |
| Postscript f Incomplete Information on Both Sides | 215 | |
| Postscript g Both Players Have Incomplete Information and Know Their Own Payoffs | 216 | |
| Postscript h Identical Information | 216 | |
| Postscript i Games Without a Recursive Structure | 219 | |
| Ch. 5 | Repeated Games of Incomplete Information: An Approach to the Non-Zero-Sum Case | |
| 2 | Equilibrium Points and Equilibrium Payoffs in a 2-Player One-Shot Game | 226 |
| 3 | Equilibrium Payoffs of an Infinitely-Repeated 2-Player Game with Complete Information | 233 |
| 4 | Repeated Games with Incomplete Information | 240 |
| 5 | The Possibility of Cheating | 257 |
| 6 | Joint Plans | 260 |
| 7 | More Equilibrium Points | 269 |
| 8 | More Equilibrium Payoffs | 277 |
| Postscript a Existence of Equilibria | 294 | |
| Postscript b The Folk Theorem and Individual Rationality | 294 | |
| Postscript c Characterization of Equilibria: Background | 294 | |
| Postscript d Characterization of Equilibria: Geometric Preliminaries | 296 | |
| Postscript e Characterization of Equilibria: Precise Formulation | 298 | |
| Postscript f Characterization of Equilibria: Outline of Hart's Proof | 301 | |
| Postscript g An Economic Example Requiring Unboundedly Many Communications | 311 | |
| Postscript h Known Own Payoffs | 320 | |
| Postscript i Communication Equilibria | 320 | |
| Postscript j Perturbations | 321 | |
| Bibliography | 323 | |
| Index | 335 |
Overview
During the height of the Cold War, between 1965 and 1968, Robert Aumann, Michael Maschler, and Richard Stearns collaborated on research on the dynamics of arms control negotiations that has since become foundational to work on repeated games. These five seminal papers are collected here for the first time, with the addition of "postscripts" describing many of the developments since the papers were written. The basic model studied throughout the book is one in which players ignorant about the game being played ...