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Mean Field Games And Mean Field Type Control Theory
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Book Synopsis Mean Field Games and Mean Field Type Control Theory by : Alain Bensoussan
Download or read book Mean Field Games and Mean Field Type Control Theory written by Alain Bensoussan and published by Springer Science & Business Media. This book was released on 2013-10-16 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mean field games and Mean field type control introduce new problems in Control Theory. The terminology “games” may be confusing. In fact they are control problems, in the sense that one is interested in a single decision maker, whom we can call the representative agent. However, these problems are not standard, since both the evolution of the state and the objective functional is influenced but terms which are not directly related to the state or the control of the decision maker. They are however, indirectly related to him, in the sense that they model a very large community of agents similar to the representative agent. All the agents behave similarly and impact the representative agent. However, because of the large number an aggregation effect takes place. The interesting consequence is that the impact of the community can be modeled by a mean field term, but when this is done, the problem is reduced to a control problem.
Book Synopsis Mean Field Games and Mean Field Type Control Theory by : Alain Bensoussan
Download or read book Mean Field Games and Mean Field Type Control Theory written by Alain Bensoussan and published by Springer. This book was released on 2013-10-17 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mean field games and Mean field type control introduce new problems in Control Theory. The terminology “games” may be confusing. In fact they are control problems, in the sense that one is interested in a single decision maker, whom we can call the representative agent. However, these problems are not standard, since both the evolution of the state and the objective functional is influenced but terms which are not directly related to the state or the control of the decision maker. They are however, indirectly related to him, in the sense that they model a very large community of agents similar to the representative agent. All the agents behave similarly and impact the representative agent. However, because of the large number an aggregation effect takes place. The interesting consequence is that the impact of the community can be modeled by a mean field term, but when this is done, the problem is reduced to a control problem.
Book Synopsis Mean Field Games and Mean Field Type Control Theory by : Alain Bensoussan
Download or read book Mean Field Games and Mean Field Type Control Theory written by Alain Bensoussan and published by Springer. This book was released on 2013-10-31 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mean field games and Mean field type control introduce new problems in Control Theory. The terminology games may be confusing. In fact they are control problems, in the sense that one is interested in a single decision maker, whom we can call the representative agent. However, these problems are not standard, since both the evolution of the state and the objective functional is influenced but terms which are not directly related to the state or the control of the decision maker. They are however, indirectly related to him, in the sense that they model a very large community of agents similar to the representative agent. All the agents behave similarly and impact the representative agent. However, because of the large number an aggregation effect takes place. The interesting consequence is that the impact of the community can be modeled by a mean field term, but when this is done, the problem is reduced to a control problem.
Book Synopsis Probabilistic Theory of Mean Field Games with Applications I by : René Carmona
Download or read book Probabilistic Theory of Mean Field Games with Applications I written by René Carmona and published by Springer. This book was released on 2018-03-01 with total page 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic mean field control problems. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.
Download or read book Mean Field Games written by Yves Achdou and published by Springer Nature. This book was released on 2021-01-19 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides an introduction to the theory of Mean Field Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a mean-field model for Nash equilibria in the strategic interaction of a large number of agents. Besides giving an accessible presentation of the main features of mean-field game theory, the volume offers an overview of recent developments which explore several important directions: from partial differential equations to stochastic analysis, from the calculus of variations to modeling and aspects related to numerical methods. Arising from the CIME Summer School "Mean Field Games" held in Cetraro in 2019, this book collects together lecture notes prepared by Y. Achdou (with M. Laurière), P. Cardaliaguet, F. Delarue, A. Porretta and F. Santambrogio. These notes will be valuable for researchers and advanced graduate students who wish to approach this theory and explore its connections with several different fields in mathematics.
Book Synopsis Mean-Field-Type Games for Engineers by : Julian Barreiro-Gomez
Download or read book Mean-Field-Type Games for Engineers written by Julian Barreiro-Gomez and published by CRC Press. This book was released on 2021-11-18 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: The contents of this book comprise an appropriate background to start working and doing research on mean-field-type control and game theory. To make the exposition and explanation even easier, we first study the deterministic optimal control and differential linear-quadratic games. Then, we progressively add complexity step-by-step and little-by-little to the problem settings until we finally study and analyze mean-field-type control and game problems incorporating several stochastic processes, e.g., Brownian motions, Poisson jumps, and random coefficients. We go beyond the Nash equilibrium, which provides a solution for non- cooperative games, by analyzing other game-theoretical concepts such as the Berge, Stackelberg, adversarial/robust, and co-opetitive equilibria. For the mean-field-type game analysis, we provide several numerical examples using a Matlab-based user-friendly toolbox that is available for the free use to the readers of this book. We present several engineering applications in both continuous and discrete time. Among these applications we find the following: water distribution systems, micro-grid energy storage, stirred tank reactor, mechanism design for evolutionary dynamics, multi-level building evacuation problem, and the COVID-19 propagation control. Julian Barreiro-Gomez Hamidou Tembine With such a demand from engineering audiences, this book is very timely and provides a thorough study of mean-field-type game theory. The strenuous protagonist of this book is to bridge between the theoretical findings and engineering solutions. The book introduces the basics first, and then mathematical frameworks are elaborately explained. The engineering application examples are shown in detail, and the popular learning approaches are also investigated. Those advantageous characteristics will make this book a comprehensive handbook of many engineering fields for many years, and I will buy one when it gets published. Zhu Han
Book Synopsis The Master Equation and the Convergence Problem in Mean Field Games by : Pierre Cardaliaguet
Download or read book The Master Equation and the Convergence Problem in Mean Field Games written by Pierre Cardaliaguet and published by Princeton University Press. This book was released on 2019-08-13 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.
Book Synopsis Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications by : Rene Carmona
Download or read book Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications written by Rene Carmona and published by SIAM. This book was released on 2016-02-18 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this textbook is to introduce students to the stochastic analysis tools that play an increasing role in the probabilistic approach to optimization problems, including stochastic control and stochastic differential games. While optimal control is taught in many graduate programs in applied mathematics and operations research, the author was intrigued by the lack of coverage of the theory of stochastic differential games. This is the first title in SIAM?s Financial Mathematics book series and is based on the author?s lecture notes. It will be helpful to students who are interested in stochastic differential equations (forward, backward, forward-backward); the probabilistic approach to stochastic control (dynamic programming and the stochastic maximum principle); and mean field games and control of McKean?Vlasov dynamics. The theory is illustrated by applications to models of systemic risk, macroeconomic growth, flocking/schooling, crowd behavior, and predatory trading, among others.
Book Synopsis Regularity Theory for Mean-Field Game Systems by : Diogo A. Gomes
Download or read book Regularity Theory for Mean-Field Game Systems written by Diogo A. Gomes and published by Springer. This book was released on 2016-09-14 with total page 165 pages. Available in PDF, EPUB and Kindle. Book excerpt: Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.
Book Synopsis Viability Theory by : Jean Pierre Aubin
Download or read book Viability Theory written by Jean Pierre Aubin and published by . This book was released on 1991 with total page 584 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work examines viability theory and its applications to control theory and differential games. The emphasis is on the construction of feedbacks and dynamical systems by myopic optimization methods. Systems of first-order partial differential inclusions, whose solutions are feedbacks, are constructed and investigated. Basic results are then extended to the case of fuzzy control problems, distributed control problems, and control systems with delays and memory. Aimed at graduate students and research mathematicians, both pure and applied, this book offers specialists in control and nonlinear systems tools to take into account general state constraints. Viability theory also allows researchers in other disciplinesâartificial intelligence, economics, game theory, theoretical biology, population genetics, cognitive sciencesâto go beyond deterministic models by studying them in a dynamical or evolutionary perspective in an uncertain environment. "The book is a compendium of the state of knowledge about viability...Mathematically, the book should be accessible to anyone who has had basic graduate courses in modern analysis and functional analysisâ¦The concepts are defined and many proofs of the requisite results are reproduced here, making the present book essentially self-contained." (Bulletin of the AMS) "Because of the wide scope, the book is an ideal reference for people encountering problems related to viability theory in their researchâ¦It gives a very thorough mathematical presentation. Very useful for anybody confronted with viability constraints." (Mededelingen van het Wiskundig Genootschap)
Book Synopsis Probabilistic Theory of Mean Field Games with Applications II by : René Carmona
Download or read book Probabilistic Theory of Mean Field Games with Applications II written by René Carmona and published by Springer. This book was released on 2018-03-08 with total page 712 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.
Book Synopsis Mean Field Games by : François Delarue
Download or read book Mean Field Games written by François Delarue and published by American Mathematical Society. This book was released on 2021-12-14 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is based on lectures delivered at the 2020 AMS Short Course “Mean Field Games: Agent Based Models to Nash Equilibria,” held January 13–14, 2020, in Denver, Colorado. Mean field game theory offers a robust methodology for studying large systems of interacting rational agents. It has been extraordinarily successful and has continued to develop since its inception. The six chapters that make up this volume provide an overview of the subject, from the foundations of the theory to applications in economics and finance, including computational aspects. The reader will find a pedagogical introduction to the main ingredients, from the forward-backward mean field game system to the master equation. Also included are two detailed chapters on the connection between finite games and mean field games, with a pedestrian description of the different methods available to solve the convergence problem. The volume concludes with two contributions on applications of mean field games and on existing numerical methods, with an opening to machine learning techniques.
Book Synopsis Optimal Control Theory by : Zhongjing Ma
Download or read book Optimal Control Theory written by Zhongjing Ma and published by Springer Nature. This book was released on 2021-01-30 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on how to implement optimal control problems via the variational method. It studies how to implement the extrema of functional by applying the variational method and covers the extrema of functional with different boundary conditions, involving multiple functions and with certain constraints etc. It gives the necessary and sufficient condition for the (continuous-time) optimal control solution via the variational method, solves the optimal control problems with different boundary conditions, analyzes the linear quadratic regulator & tracking problems respectively in detail, and provides the solution of optimal control problems with state constraints by applying the Pontryagin’s minimum principle which is developed based upon the calculus of variations. And the developed results are applied to implement several classes of popular optimal control problems and say minimum-time, minimum-fuel and minimum-energy problems and so on. As another key branch of optimal control methods, it also presents how to solve the optimal control problems via dynamic programming and discusses the relationship between the variational method and dynamic programming for comparison. Concerning the system involving individual agents, it is also worth to study how to implement the decentralized solution for the underlying optimal control problems in the framework of differential games. The equilibrium is implemented by applying both Pontryagin’s minimum principle and dynamic programming. The book also analyzes the discrete-time version for all the above materials as well since the discrete-time optimal control problems are very popular in many fields.
Book Synopsis Advances in Dynamic and Mean Field Games by : Joseph Apaloo
Download or read book Advances in Dynamic and Mean Field Games written by Joseph Apaloo and published by Birkhäuser. This book was released on 2018-01-03 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: This contributed volume considers recent advances in dynamic games and their applications, based on presentations given at the 17th Symposium of the International Society of Dynamic Games, held July 12-15, 2016, in Urbino, Italy. Written by experts in their respective disciplines, these papers cover various aspects of dynamic game theory including mean-field games, stochastic and pursuit-evasion games, and computational methods for dynamic games. Topics covered include Pedestrian flow in crowded environments Models for climate change negotiations Nash Equilibria for dynamic games involving Volterra integral equations Differential games in healthcare markets Linear-quadratic Gaussian dynamic games Aircraft control in wind shear conditions Advances in Dynamic and Mean-Field Games presents state-of-the-art research in a wide spectrum of areas. As such, it serves as a testament to the continued vitality and growth of the field of dynamic games and their applications. It will be of interest to an interdisciplinary audience of researchers, practitioners, and graduate students.
Book Synopsis The Master Equation and the Convergence Problem in Mean Field Games by : Pierre Cardaliaguet
Download or read book The Master Equation and the Convergence Problem in Mean Field Games written by Pierre Cardaliaguet and published by Princeton University Press. This book was released on 2019-08-13 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.
Book Synopsis Stochastic Games and Applications by : Abraham Neyman
Download or read book Stochastic Games and Applications written by Abraham Neyman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is based on lectures given at the NATO Advanced Study Institute on "Stochastic Games and Applications," which took place at Stony Brook, NY, USA, July 1999. It gives the editors great pleasure to present it on the occasion of L.S. Shapley's eightieth birthday, and on the fiftieth "birthday" of his seminal paper "Stochastic Games," with which this volume opens. We wish to thank NATO for the grant that made the Institute and this volume possible, and the Center for Game Theory in Economics of the State University of New York at Stony Brook for hosting this event. We also wish to thank the Hebrew University of Jerusalem, Israel, for providing continuing financial support, without which this project would never have been completed. In particular, we are grateful to our editorial assistant Mike Borns, whose work has been indispensable. We also would like to acknowledge the support of the Ecole Poly tech nique, Paris, and the Israel Science Foundation. March 2003 Abraham Neyman and Sylvain Sorin ix STOCHASTIC GAMES L.S. SHAPLEY University of California at Los Angeles Los Angeles, USA 1. Introduction In a stochastic game the play proceeds by steps from position to position, according to transition probabilities controlled jointly by the two players.
Book Synopsis Stochastic Linear-Quadratic Optimal Control Theory: Differential Games and Mean-Field Problems by : Jingrui Sun
Download or read book Stochastic Linear-Quadratic Optimal Control Theory: Differential Games and Mean-Field Problems written by Jingrui Sun and published by Springer Nature. This book was released on 2020-06-29 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gathers the most essential results, including recent ones, on linear-quadratic optimal control problems, which represent an important aspect of stochastic control. It presents results for two-player differential games and mean-field optimal control problems in the context of finite and infinite horizon problems, and discusses a number of new and interesting issues. Further, the book identifies, for the first time, the interconnections between the existence of open-loop and closed-loop Nash equilibria, solvability of the optimality system, and solvability of the associated Riccati equation, and also explores the open-loop solvability of mean-filed linear-quadratic optimal control problems. Although the content is largely self-contained, readers should have a basic grasp of linear algebra, functional analysis and stochastic ordinary differential equations. The book is mainly intended for senior undergraduate and graduate students majoring in applied mathematics who are interested in stochastic control theory. However, it will also appeal to researchers in other related areas, such as engineering, management, finance/economics and the social sciences.
