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This book explores the solution of dynamical games problems under uncertainty by means of the Bellman-Lyapunov function. The author first describes the foundations of differential games under uncertainties and presents examples from economic dynamics needed for the investigation. He focuses on the notion of the vector guarantee, its properties, and the methods of construction. Part Two explores differential linear quadratic games under uncertainty. Here the author proposes new guaranteeing solutions based on the concept of the equilibrium of objections and counter-objections as well as the active equilibrium. Each chapter includes exercises, and solutions are provided at the end of the book.
List of contents
The Simplist Concepts and Examples. Some Concepts in the Theory of Differential Games Under Uncertainty. Game Problems in Mechanical and Economical Systems. Vector-Valued Guarantees. Vector-Valued Guarantees Can Exist or Not. Converse Problem. Equilibrium of Nash Under Uncertainty. Equilibrium of Threats and Counterthreats Under Uncertainty. Singularities of the Nash Equilibrium. Formalization and Properties Unimprovable Equilibriums. Comparison with Nash Equilibrium. Formalization of Unimprovable Equilibriums in Differential Game. Auxiliary Propositions. Sufficient Conditions for the Saddle-Point Analogy. Unimprovable Guaranteeing Equilibriums (Vector-Valued Max/min Analogy). Active Equilibrium Under Uncertainty. Berge Equilibrium Under Uncertainty. Formalization of the Solutions. Games with Separable Payoff Function. Strictly Convex Games Under Uncertainty. Properties of Berge Equilibrium. Linear-Quadratic Differential Game of Three Persons Under Uncertainty. Appendix 1: From the Theory of Differential Equations. Appendix 2: From the Theory of Quadratic Forms. Appendix 3: From the Theory of Mathematical Programming. Appendix 4: Auxiliary Propositions.
About the author
Vladislav I Zhukovskiy
Summary
In this text, coefficient criteria are derived for numerous new and relevant problems in the theory of linear - quadratic multi-player differential games in cases when: the player formulate their strategies independently (non co-operative games) and use non-Nash equilibria.
