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Stochastic differential games represent one of the most complex forms of decision making under uncertainty. In particular, interactions between strategic behaviors, dynamic evolution and stochastic elements have to be considered simultaneously. The complexity of stochastic differential games generally leads to great difficulties in the derivation of solutions. Cooperative games hold out the promise of more socially optimal and group efficient solutions to problems involving strategic actions. Despite urgent calls for national and international cooperation, the absence of formal solutions has precluded rigorous analysis of this problem. The book supplies effective tools for rigorous study of cooperative stochastic differential games. In particular, a generalized theorem for the derivation of analytically tractable "payoff distribution procedure" of subgame consistent solution is presented. Being capable of deriving analytical tractable solutions, the work is not only theoretically interesting but would enable the hitherto intractable problems in cooperative stochastic differential games to be fruitfully explored. Currently, this book is the first ever volume devoted to cooperative stochastic differential games. It aims to provide its readers an effective tool to analyze cooperative arrangements of conflict situations with uncertainty over time. Cooperative game theory has succeeded in offering many applications of game theory in operations research, management, economics, politics and other disciplines. The extension of these applications to a dynamic environment with stochastic elements should be fruitful. The book will be of interest to game theorists, mathematicians, economists, policy-makers, corporate planners and graduate students

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Deterministic and Stochastic Differential Games . . . . . . . . . . . 7
2.1 Dynamic Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Stochastic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Differential Games and their Solution Concepts . . . . . . . . . . . . . 22
2.2.1 Open-loop Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Closed-loop Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Feedback Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Application of Differential Games in Economics . . . . . . . . . . . . . 29
2.3.1 Open-loop Solution in Competitive Advertising . . . . . . . 29
2.3.2 Feedback Solution in Competitive Advertising. . . . . . . . . 31
2.4 Infinite-Horizon Differential Games . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 Game Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.2 Infinite-Horizon Duopolistic Competition . . . . . . . . . . . . . 36
2.5 Stochastic Differential Games and their Solutions . . . . . . . . . . . . 38
2.6 An Application of Stochastic Differential Games in Resource
Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.7 Infinite-Horizon Stochastic Differential Games . . . . . . . . . . . . . . . 43
2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Cooperative Differential Games in Characteristic
Function Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1 Cooperative Differential Games in Characteristic Function
Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Game Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Solution Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Imputation in a Dynamic Context . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Principle of Dynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Dynamic Stable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Payoff Distribution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6 An Analysis in Pollution Control . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6.1 Decomposition Over Time of the Shapley Value . . . . . . . 63
3.6.2 A Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6.3 Rationale for the Algorithm and the Special
Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Illustration with Specific Functional Forms . . . . . . . . . . . . . . . . . 69
3.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4 Two-person Cooperative Differential Games with
Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1 Game Formulation and Noncooperative Outcome . . . . . . . . . . . . 75
4.2 Cooperative Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.1 Group Rationality and Optimal Trajectory . . . . . . . . . . . 80
4.2.2 Individual Rationality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Dynamically Stable Cooperation and the Notion of Time
Consistency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Equilibrating Transitory Compensation . . . . . . . . . . . . . . . . . . . . 88
4.4.1 Time Consistent Payoff Distribution Procedures . . . . . . . 88
4.4.2 Time Consistent Solutions under Specific Optimality
Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5 An Illustration in Cooperative Resource Extraction . . . . . . . . . . 91
4.6 An Economic Exegesis of Transitory Compensations . . . . . . . . . 93
4.7 Infinite-Horizon Cooperative Differential Games . . . . . . . . . . . . . 94
4.8 Games with Nontransferable Payoffs . . . . . . . . . . . . . . . . . . . . . . . 101
4.8.1 Pareto Optimal Trajectories under Cooperation . . . . . . . 102
4.8.2 Individual Player’s Payoffs under Cooperation . . . . . . . . 104
4.8.3 Time Consistent Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.8.4 An Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.8.5 A Proposed Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.9 Appendix to Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Two-person Cooperative Stochastic Differential Games . . . . 121
5.1 Game Formulation and Noncooperative Outcome . . . . . . . . . . . . 121
5.2 Cooperative Arrangement under Uncertainty . . . . . . . . . . . . . . . . 126
5.2.1 Group Rationality and Optimal Trajectory . . . . . . . . . . . 126
5.2.2 Individual Rationality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 Dynamically Stable Cooperation and the Notion of Subgame
Consistency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.4 Transitory Compensation and Payoff Distribution Procedures . 135
5.5 Subgame Consistent Solutions under Specific Optimality
Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.5.1 The Nash Bargaining/Shapley Value Solution . . . . . . . . . 137
5.5.2 Proportional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.6 An Application in Cooperative Resource Extraction under
Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.7 An Exegesis of Transitory Compensation under Uncertainty . . 142
5.8 Infinite-Horizon Cooperative Stochastic Differential Games . . . 144
5.9 Appendix to Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6 Multiplayer Cooperative Stochastic Differential Games . . . . 157
6.1 A Class of Multiplayer Games in Cooperative Technology
Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2 Theoretical Underpinning in a Deterministic Framework . . . . . . 158
6.2.1 A Dynamic Model of Joint Venture . . . . . . . . . . . . . . . . . . 158
6.2.2 Coalition Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2.3 The Dynamic Shapley Value Imputation . . . . . . . . . . . . . 161
6.2.4 Transitory Compensation. . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.3 An Application in Joint Venture . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.4 The Stochastic Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.4.1 Expected Coalition Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.4.2 Stochastic Dynamic Shapley Value . . . . . . . . . . . . . . . . . . 175
6.4.3 Transitory Compensation. . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.5 An Application in Cooperative R&D under Uncertainty . . . . . . 181
6.6 Infinite-Horizon Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.7 An Example of Infinite-Horizon Joint Venture . . . . . . . . . . . . . . . 190
6.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7 Cooperative Stochastic Differential Games with
Nontransferable Payoffs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.1 Game Formulation and Noncooperative Outcome . . . . . . . . . . . . 199
7.2 Cooperative Arrangement under Uncertainty and
Nontransferable Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.2.1 Pareto Optimal Trajectories . . . . . . . . . . . . . . . . . . . . . . . . 203
7.3 Individual Player’s Payoffs under Cooperation . . . . . . . . . . . . . . . 207
7.4 Subgame Consistent Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.5 A Proposed Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.5.1 Typical Configurations of St . . . . . . . . . . . . . . . . . . . . . . . . 216
7.5.2 A Subgame Consistent Solution . . . . . . . . . . . . . . . . . . . . . 217
7.5.3 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7.6 Infinite-Horizon Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
7.6.1 Noncooperative Equilibria and Pareto Optimal
Trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
· · · · · · (收起)