DIFFERENTIAL-NONDIFFERENTIABLE GAMES AND APPLICATIONS IN ECONOMICS / Rania Benmenni

Public ISBD Titre : DIFFERENTIAL-NONDIFFERENTIABLE GAMES AND APPLICATIONS IN ECONOMICS Type de document : document électronique Auteurs : Rania Benmenni, Auteur ; Nourreddine Daili, Directeur de thèse Editeur : Setif:UFA Année de publication : 2025 Importance : 1 vol (60 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Nonzero-sum differential games Maximum principle Dynamic programming principle Viscosity solutions Index. décimale : 510 - Mathématique Résumé : The main objective of this thesis is to present the connection between the adjoint variables in the maximum principle (MP) and the value function in the dynamic programming principle (DPP) for two-player nonzero-sum differential games, both in the smooth and nonsmooth cases. This relationship is established in terms of derivatives in the smooth case and through viscosity solutions when the value function is not smooth, with economic interpretations related to the adjoint variables. In the second part, we apply a numerical method based on the Jacobi spectral method (JSM) to solve the nonlinear two-point boundary value problems (TPBVPs) derived from the maximum principle. These problems are then transferred into a system of algebraic equations to determine the open-loop Nash equilibrium (OLNE) for nonzero-sum differential games. Illustrative examples are presented to demonstrate the effectiveness and validity of the proposed method. Note de contenu : Sommaire 1 An Overview of Differential Games Theory 4 1.1 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Approaches to Solving Optimal Control Problem . . . . . . . . . . . . . . 6 1.2 Basic Notions of Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Game Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Strategic-Form Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Differential Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Information Structures and Strategies . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Linear Quadratic Differential Games (LQDGs) . . . . . . . . . . . . . . . . 14 2 A Connection Between the Adjoint Variables and Value Function for Differential Games 16 2.1 Formulation of the Game Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 Maximum Principle (MP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.2 Dynamic Programming Principle (DPP) . . . . . . . . . . . . . . . . . . . . 19 2.2 The Connection Between MP and DPP : Smooth Case . . . . . . . . . . . . . . . . 21 2.3 The Connection Between MP and DPP : Nonsmooth Case . . . . . . . . . . . . . . 23 3 Applications To Economic 30 3.1 Producer-Consumer Game with Sticky Price . . . . . . . . . . . . . . . . . . . . . 30 3.1.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 The Connection Between MP and DPP . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1 Smooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.2 Nonsmooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Jacobi Spectral Method for Solving Differential Games 37 4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Jacobi Spectral Method for Nonzero-sum Differential Games . . . . . . . . . . . . 39 4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Conclusion 56 Bibliography 60 Côte titre : DM/0212 DIFFERENTIAL-NONDIFFERENTIABLE GAMES AND APPLICATIONS IN ECONOMICS [document électronique] / Rania Benmenni, Auteur ; Nourreddine Daili, Directeur de thèse . - [S.l.] : Setif:UFA, 2025 . - 1 vol (60 f.) ; 29 cm. Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Nonzero-sum differential games Maximum principle Dynamic programming principle Viscosity solutions Index. décimale : 510 - Mathématique Résumé : The main objective of this thesis is to present the connection between the adjoint variables in the maximum principle (MP) and the value function in the dynamic programming principle (DPP) for two-player nonzero-sum differential games, both in the smooth and nonsmooth cases. This relationship is established in terms of derivatives in the smooth case and through viscosity solutions when the value function is not smooth, with economic interpretations related to the adjoint variables. In the second part, we apply a numerical method based on the Jacobi spectral method (JSM) to solve the nonlinear two-point boundary value problems (TPBVPs) derived from the maximum principle. These problems are then transferred into a system of algebraic equations to determine the open-loop Nash equilibrium (OLNE) for nonzero-sum differential games. Illustrative examples are presented to demonstrate the effectiveness and validity of the proposed method. Note de contenu : Sommaire 1 An Overview of Differential Games Theory 4 1.1 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Approaches to Solving Optimal Control Problem . . . . . . . . . . . . . . 6 1.2 Basic Notions of Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Game Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Strategic-Form Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Differential Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Information Structures and Strategies . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Linear Quadratic Differential Games (LQDGs) . . . . . . . . . . . . . . . . 14 2 A Connection Between the Adjoint Variables and Value Function for Differential Games 16 2.1 Formulation of the Game Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 Maximum Principle (MP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.2 Dynamic Programming Principle (DPP) . . . . . . . . . . . . . . . . . . . . 19 2.2 The Connection Between MP and DPP : Smooth Case . . . . . . . . . . . . . . . . 21 2.3 The Connection Between MP and DPP : Nonsmooth Case . . . . . . . . . . . . . . 23 3 Applications To Economic 30 3.1 Producer-Consumer Game with Sticky Price . . . . . . . . . . . . . . . . . . . . . 30 3.1.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 The Connection Between MP and DPP . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1 Smooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.2 Nonsmooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Jacobi Spectral Method for Solving Differential Games 37 4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Jacobi Spectral Method for Nonzero-sum Differential Games . . . . . . . . . . . . 39 4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Conclusion 56 Bibliography 60 Côte titre : DM/0212

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DIFFERENTIAL-NONDIFFERENTIABLE GAMES AND APPLICATIONS IN ECONOMICS / Rania Benmenni

Public ISBD Titre : DIFFERENTIAL-NONDIFFERENTIABLE GAMES AND APPLICATIONS IN ECONOMICS Type de document : document électronique Auteurs : Rania Benmenni, Auteur ; Nourreddine Daili, Directeur de thèse Editeur : Setif:UFA Année de publication : 2025 Importance : 1 vol (60 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Nonzero-sum differential games Maximum principle Dynamic programming principle Viscosity solutions Index. décimale : 510 - Mathématique Résumé : The main objective of this thesis is to present the connection between the adjoint variables in the maximum principle (MP) and the value function in the dynamic programming principle (DPP) for two-player nonzero-sum differential games, both in the smooth and nonsmooth cases. This relationship is established in terms of derivatives in the smooth case and through viscosity solutions when the value function is not smooth, with economic interpretations related to the adjoint variables. In the second part, we apply a numerical method based on the Jacobi spectral method (JSM) to solve the nonlinear two-point boundary value problems (TPBVPs) derived from the maximum principle. These problems are then transferred into a system of algebraic equations to determine the open-loop Nash equilibrium (OLNE) for nonzero-sum differential games. Illustrative examples are presented to demonstrate the effectiveness and validity of the proposed method. Note de contenu : Sommaire 1 An Overview of Differential Games Theory 4 1.1 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Approaches to Solving Optimal Control Problem . . . . . . . . . . . . . . 6 1.2 Basic Notions of Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Game Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Strategic-Form Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Differential Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Information Structures and Strategies . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Linear Quadratic Differential Games (LQDGs) . . . . . . . . . . . . . . . . 14 2 A Connection Between the Adjoint Variables and Value Function for Differential Games 16 2.1 Formulation of the Game Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 Maximum Principle (MP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.2 Dynamic Programming Principle (DPP) . . . . . . . . . . . . . . . . . . . . 19 2.2 The Connection Between MP and DPP : Smooth Case . . . . . . . . . . . . . . . . 21 2.3 The Connection Between MP and DPP : Nonsmooth Case . . . . . . . . . . . . . . 23 3 Applications To Economic 30 3.1 Producer-Consumer Game with Sticky Price . . . . . . . . . . . . . . . . . . . . . 30 3.1.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 The Connection Between MP and DPP . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1 Smooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.2 Nonsmooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Jacobi Spectral Method for Solving Differential Games 37 4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Jacobi Spectral Method for Nonzero-sum Differential Games . . . . . . . . . . . . 39 4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Conclusion 56 Bibliography 60 Côte titre : DM/0212 DIFFERENTIAL-NONDIFFERENTIABLE GAMES AND APPLICATIONS IN ECONOMICS [document électronique] / Rania Benmenni, Auteur ; Nourreddine Daili, Directeur de thèse . - [S.l.] : Setif:UFA, 2025 . - 1 vol (60 f.) ; 29 cm. Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Nonzero-sum differential games Maximum principle Dynamic programming principle Viscosity solutions Index. décimale : 510 - Mathématique Résumé : The main objective of this thesis is to present the connection between the adjoint variables in the maximum principle (MP) and the value function in the dynamic programming principle (DPP) for two-player nonzero-sum differential games, both in the smooth and nonsmooth cases. This relationship is established in terms of derivatives in the smooth case and through viscosity solutions when the value function is not smooth, with economic interpretations related to the adjoint variables. In the second part, we apply a numerical method based on the Jacobi spectral method (JSM) to solve the nonlinear two-point boundary value problems (TPBVPs) derived from the maximum principle. These problems are then transferred into a system of algebraic equations to determine the open-loop Nash equilibrium (OLNE) for nonzero-sum differential games. Illustrative examples are presented to demonstrate the effectiveness and validity of the proposed method. Note de contenu : Sommaire 1 An Overview of Differential Games Theory 4 1.1 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Approaches to Solving Optimal Control Problem . . . . . . . . . . . . . . 6 1.2 Basic Notions of Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Game Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Strategic-Form Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Differential Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Information Structures and Strategies . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Linear Quadratic Differential Games (LQDGs) . . . . . . . . . . . . . . . . 14 2 A Connection Between the Adjoint Variables and Value Function for Differential Games 16 2.1 Formulation of the Game Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 Maximum Principle (MP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.2 Dynamic Programming Principle (DPP) . . . . . . . . . . . . . . . . . . . . 19 2.2 The Connection Between MP and DPP : Smooth Case . . . . . . . . . . . . . . . . 21 2.3 The Connection Between MP and DPP : Nonsmooth Case . . . . . . . . . . . . . . 23 3 Applications To Economic 30 3.1 Producer-Consumer Game with Sticky Price . . . . . . . . . . . . . . . . . . . . . 30 3.1.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 The Connection Between MP and DPP . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1 Smooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.2 Nonsmooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Jacobi Spectral Method for Solving Differential Games 37 4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Jacobi Spectral Method for Nonzero-sum Differential Games . . . . . . . . . . . . 39 4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Conclusion 56 Bibliography 60 Côte titre : DM/0212

Exemplaires (1)

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DM/0212	DM/0212	Thèse	Bibliothèque des sciences	Anglais	Disponible Disponible

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