Thomas A. Weber
- Published in print:
- 2011
- Published Online:
- August 2013
- ISBN:
- 9780262015738
- eISBN:
- 9780262298483
- Item type:
- chapter
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262015738.003.0004
- Subject:
- Mathematics, Probability / Statistics
This chapter provides an introduction to games in discrete time with complete and incomplete information. Based on this body of classical game theory, it goes on to discuss continuous-time ...
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This chapter provides an introduction to games in discrete time with complete and incomplete information. Based on this body of classical game theory, it goes on to discuss continuous-time differential games.Less
This chapter provides an introduction to games in discrete time with complete and incomplete information. Based on this body of classical game theory, it goes on to discuss continuous-time differential games.
João P. Hespanha
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175218
- eISBN:
- 9781400885442
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175218.003.0014
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter discusses a new class of games known as dynamic games. It begins by considering a two-player multi-stage game in extensive form in which the overall tree structure can be mathematically ...
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This chapter discusses a new class of games known as dynamic games. It begins by considering a two-player multi-stage game in extensive form in which the overall tree structure can be mathematically described in a manner that actually allows for games that are more general than those typically described in extensive form. These include games described by graphs that are not trees, games with infinitely many stages, and games with action spaces that are not finite sets. The equation for solving dynamic games is often called the dynamics of the game. The chapter also examines the information structures of dynamic games and concludes with an overview of continuous time differential games and differential games with variable termination time.Less
This chapter discusses a new class of games known as dynamic games. It begins by considering a two-player multi-stage game in extensive form in which the overall tree structure can be mathematically described in a manner that actually allows for games that are more general than those typically described in extensive form. These include games described by graphs that are not trees, games with infinitely many stages, and games with action spaces that are not finite sets. The equation for solving dynamic games is often called the dynamics of the game. The chapter also examines the information structures of dynamic games and concludes with an overview of continuous time differential games and differential games with variable termination time.
Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions
- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691190716
- eISBN:
- 9780691193717
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691190716.001.0001
- Subject:
- Mathematics, Applied Mathematics
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 ...
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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. The 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. The 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.Less
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. The 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. The 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.
Anastasios Xepapadeas
- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199692873
- eISBN:
- 9780191738371
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199692873.003.0006
- Subject:
- Economics and Finance, Development, Growth, and Environmental
Robust policy rules and precaution might be called for, under conditions of scientific uncertainty, when a policy maker is concerned about possible misspecification of the natural system that is used ...
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Robust policy rules and precaution might be called for, under conditions of scientific uncertainty, when a policy maker is concerned about possible misspecification of the natural system that is used to model pollution dynamics. Precaution, however, could be costly. The present chapter develops a conceptual framework for designing robust policy rules and estimating the cost of being precautious in the context of an international pollution control problem. Cooperative and non-cooperative robust policy rules are determined and the cost, in terms of value loss, of being robust relative to conventional policy rules is estimated.Less
Robust policy rules and precaution might be called for, under conditions of scientific uncertainty, when a policy maker is concerned about possible misspecification of the natural system that is used to model pollution dynamics. Precaution, however, could be costly. The present chapter develops a conceptual framework for designing robust policy rules and estimating the cost of being precautious in the context of an international pollution control problem. Cooperative and non-cooperative robust policy rules are determined and the cost, in terms of value loss, of being robust relative to conventional policy rules is estimated.
Thomas A. Weber
- Published in print:
- 2011
- Published Online:
- August 2013
- ISBN:
- 9780262015738
- eISBN:
- 9780262298483
- Item type:
- book
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262015738.001.0001
- Subject:
- Mathematics, Probability / Statistics
This book bridges optimal control theory and economics, discussing ordinary differential equations (ODEs), optimal control, game theory, and mechanism design in one volume. Technically rigorous and ...
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This book bridges optimal control theory and economics, discussing ordinary differential equations (ODEs), optimal control, game theory, and mechanism design in one volume. Technically rigorous and largely self-contained, it provides an introduction to the use of optimal control theory for deterministic continuous-time systems in economics. The theory of ordinary differential equations is the backbone of the theory developed in the book, and Chapter 2 offers a detailed review of basic concepts in the theory of ODEs, including the solution of systems of linear ODEs, state-space analysis, potential functions, and stability analysis. Following this, the book covers the main results of optimal control theory, in particular necessary and sufficient optimality conditions; game theory, with an emphasis on differential games; and the application of control-theoretic concepts to the design of economic mechanisms. Appendices provide a mathematical review and full solutions to all end-of-chapter problems. The material is presented at three levels: single-person decision making; games, in which a group of decision makers interact strategically; and mechanism design, which is concerned with a designer’s creation of an environment in which players interact to maximize the designer’s objective. The book focuses on applications; the problems are an integral part of the text.Less
This book bridges optimal control theory and economics, discussing ordinary differential equations (ODEs), optimal control, game theory, and mechanism design in one volume. Technically rigorous and largely self-contained, it provides an introduction to the use of optimal control theory for deterministic continuous-time systems in economics. The theory of ordinary differential equations is the backbone of the theory developed in the book, and Chapter 2 offers a detailed review of basic concepts in the theory of ODEs, including the solution of systems of linear ODEs, state-space analysis, potential functions, and stability analysis. Following this, the book covers the main results of optimal control theory, in particular necessary and sufficient optimality conditions; game theory, with an emphasis on differential games; and the application of control-theoretic concepts to the design of economic mechanisms. Appendices provide a mathematical review and full solutions to all end-of-chapter problems. The material is presented at three levels: single-person decision making; games, in which a group of decision makers interact strategically; and mechanism design, which is concerned with a designer’s creation of an environment in which players interact to maximize the designer’s objective. The book focuses on applications; the problems are an integral part of the text.
Darko Jus and Volker Meier
- Published in print:
- 2014
- Published Online:
- January 2015
- ISBN:
- 9780262027885
- eISBN:
- 9780262319836
- Item type:
- chapter
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262027885.003.0009
- Subject:
- Economics and Finance, Development, Growth, and Environmental
Employing a Stackelberg differential game approach, the chapter derives the carbon tax chosen by a climate coalition of resource consuming countries which purchase the fossil resource from a ...
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Employing a Stackelberg differential game approach, the chapter derives the carbon tax chosen by a climate coalition of resource consuming countries which purchase the fossil resource from a representative competitive resource supplier. The global climate coalition reduces the speed at which the global fossil resource stock is depleted over time to the socially efficient level by levying the Pigou tax on resource consumption. If the climate coalition is incomplete, the chosen unilateral carbon tax falls short of the Pigou tax. Furthermore, international carbon leakage undermines the effectiveness of the unilateral carbon tax in slowing down the speed of global resource extraction. Nevertheless, under the assumptions made, also the incomplete climate coalition is able to slow down the speed of global extraction to some extent because the chosen carbon tax is time-consistent, irrespective of whether the coalition is global or incomplete.Less
Employing a Stackelberg differential game approach, the chapter derives the carbon tax chosen by a climate coalition of resource consuming countries which purchase the fossil resource from a representative competitive resource supplier. The global climate coalition reduces the speed at which the global fossil resource stock is depleted over time to the socially efficient level by levying the Pigou tax on resource consumption. If the climate coalition is incomplete, the chosen unilateral carbon tax falls short of the Pigou tax. Furthermore, international carbon leakage undermines the effectiveness of the unilateral carbon tax in slowing down the speed of global resource extraction. Nevertheless, under the assumptions made, also the incomplete climate coalition is able to slow down the speed of global extraction to some extent because the chosen carbon tax is time-consistent, irrespective of whether the coalition is global or incomplete.