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Differential Games

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  • First Online:
Complex Social and Behavioral Systems

Part of the book series: Encyclopedia of Complexity and Systems Science Series ((ECSSS))

  • 914 Accesses

  • Originally published in
  • R. A. Meyers (ed.), Encyclopedia of Complexity and Systems Science, © Springer Science+Business Media LLC 2018

Glossary

Dynamics:

This is the law which governs the evolution of the system: for differential games it is a differential equation.

Strategies:

This is the way a player chooses his control as a function of the state of the system and of the action of his opponents.

Information:

This is the set of parameters known by the player in order to build his strategy.

Definition of the Subject and Its Importance

Differential games is a mathematical theory which concerns with problems of conflicts modelized as game problems in which the state of the players depends on the time in a continuous way. The positions of the players are solution to differential equations. Differential games can be described from two different points of views, depending mainly of the field of applications. Firstly, they can be considered as games where the time is continuous. This aspect is often considered for applications in economics or management sciences. Secondly, they also can be viewed as control problems with...

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Bibliography

Primary Literature

  • Aubin J-P (1991) Viability theory. Birkhäuser, Boston

    MATH  Google Scholar 

  • Aumann RJ, Maschler M (1995) Repeated games with incomplete information. MIT Press, Cambridge

    MATH  Google Scholar 

  • Bardi M, Capuzzo Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems and control: foundations and applications. Birkhäuser, Boston. xvii, 570 pp

    Google Scholar 

  • Basar T, Bernhard P (1995) H∞-optimal control and related minimax design problems. In: A dynamic game approach, 2nd edn. Systems and control: foundations and applications. Birkhäuser, Boston

    Google Scholar 

  • Berkovitz LD (1994) A theory of differential games. In: Basar T et al (ed) Advances in dynamic games and applications. Birkhäuser, Boston. Ann Int Soc Dyn Games 1:3–22

    Google Scholar 

  • Bernhard P (1988) Differential games. In: Singh MG (ed) Systems and control encyclopedia, theory technology application. Pergamon Press, Oxford

    Google Scholar 

  • Bettiol P, Cardaliaguet P, Quincampoix M (2006) Zero-sum state constrained differential games: existence of value for Bolza problem. Int J Game Theory 34(4):495–527

    MathSciNet  MATH  Google Scholar 

  • Breakwell JV (1977) Zero-sum differential games with terminal payoff. In: Hagedorn P, Knobloch HW, Olsder GH (eds) Differential game and applications. Lecture notes in control and information sciences, vol 3. Springer, Berlin

    Google Scholar 

  • Bressan A (2011) Noncooperative differential games. Milan J Math 79:357–427

    MathSciNet  MATH  Google Scholar 

  • Buckdahn R, Cardaliaguet P, Rainer C (2004) Nash equilibrium payoffs for nonzero-sum stochastic differential games. SIAM J Control Optim 43(2):624–642

    MathSciNet  MATH  Google Scholar 

  • Buckdahn R, Djehiche B, Li J, Peng S (2009a) Mean-field backward stochastic differential equations: a limit approach. Ann Probab 37:1524–1565

    MathSciNet  MATH  Google Scholar 

  • Buckdahn R, Li J, Peng S (2009b) Mean-field backward stochastic differential equations and related partial differential equations. Stoch Process Appl 119:3133–3154

    MathSciNet  MATH  Google Scholar 

  • Buckdahn R, Li J, Quincampoix M (2013) Value function of differential games without Isaacs conditions. An approach with nonanticipative mixed strategies. Int J Game Theory 42(4):989–1020

    MathSciNet  MATH  Google Scholar 

  • Buckdahn R, Li J, Quincampoix M (2014) Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition. Ann Probab 42(4):1724–1768

    MathSciNet  MATH  Google Scholar 

  • Buckdahn R, Quincampoix M, Rainer C, Xu Y (2016) Differential games with asymmetric information and without Isaacs’ condition. Int J Game Theory 45(4):795–816

    MathSciNet  MATH  Google Scholar 

  • Cardaliaguet P (1996) A differential game with two players and one target. SIAM J Control Optim 34(4):1441–1460

    MathSciNet  MATH  Google Scholar 

  • Cardaliaguet P (1997) Nonsmooth semi-permeable barriers, Isaacs’ equation, and application to a differential game with one target and two players. Appl Math Optim 36:125–146

    MathSciNet  MATH  Google Scholar 

  • Cardaliaguet P (2007) Differential games with asymmetric information. SIAM J Control Optim 46(3):816–838

    MathSciNet  MATH  Google Scholar 

  • Cardaliaguet P (2017) The convergence problem in mean field games with local coupling. Appl Math Optim 76(1):177–215

    MathSciNet  MATH  Google Scholar 

  • Cardaliaguet P, Plasckacz S (2003) Existence and uniqueness of a Nash equilibrium feedback for a simple non zero-sum differential game. Int J Game Theory 32(4):561–593

    Google Scholar 

  • Cardaliaguet P, Quincampoix M (2008) Determinist differential games under probability knowledge of initial condition. Int Game Theory Rev 10(1):1–16

    MathSciNet  MATH  Google Scholar 

  • Cardaliaguet P, Rainer C (2009) Cardaliaguet, Pierre; Rainer, Catherine stochastic differential games with asymmetric information. Appl Math Optim 59(1):136

    Google Scholar 

  • Cardaliaguet P, Rainer C (2013) Pathwise strategies for stochastic differential games. Appl Math Optim 68(1):75–84

    MathSciNet  MATH  Google Scholar 

  • Cardaliaguet P, Quincampoix M, Saint-Pierre P (1999) Set-valued numerical analysis for optimal control and differential games. In: Bardi M, Parthasaranthy T, TES R (eds) Numerical methods for optimal control and numerical games. Annals of International Society of Dynamical Games. Birkhäuser, Boston, pp 177–249

    MATH  Google Scholar 

  • Cardaliaguet P, Quincampoix M, Saint-Pierre P (2001) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615–1632

    MathSciNet  MATH  Google Scholar 

  • Cardaliaguet P, Jimenez C, Quincampoix M (2014) Pure and random strategies in differential game with incomplete informations. J Dyn Games 1(3):363–375

    MathSciNet  MATH  Google Scholar 

  • Elliot N, Kalton N (1972) The existence of value in differential games. Mems Am Math Soc 126

    Google Scholar 

  • Evans LC, Souganidis PE (1984) Differential games and representation formulas for solutions of Hamilton-Jacobi equations. Indiana Univ Math J 282:487–502

    MathSciNet  MATH  Google Scholar 

  • Fleming W, Souganidis P (1989) On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ Math J 38(2):293–314

    MathSciNet  MATH  Google Scholar 

  • Flynn J (1973) Lion and man: the boundary constraints. SIAM J Control 11:397

    MathSciNet  MATH  Google Scholar 

  • Hamadene S, Lepeltier J-P (1995) Backward equations, stochastic control and zero-sum stochastic differential games. Stoch Int J Probab Stoch Process 54:221–231

    MathSciNet  MATH  Google Scholar 

  • Jimenez C, Quincampoix M (2018) Hamilton Jacobi Isaacs equations for differential games with asymmetric information on probabilistic initial condition. J Math Anal Appl 457(2):1422–1451

    MathSciNet  MATH  Google Scholar 

  • Klejmenov AF (1993) Nonantagonistic positional differential games. Nauka, Ekaterinburg

    MATH  Google Scholar 

  • Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2(1):229–260

    MathSciNet  MATH  Google Scholar 

  • Marigonda A, Quincampoix M (2018) Mayer control problem with probabilistic uncertainty on initial positions. J Differ Equ 264(5):3212–3252

    MathSciNet  MATH  Google Scholar 

  • Petrosjan LA (2004) Cooperation in games with incomplete information. In: Nonlinear analysis and convex analysis. Yokohama Publishers, Yokohama, pp 469–479

    Google Scholar 

  • Plaskacz S, Quincampoix M (2000) Discontinuous Mayer control problem under state-constraints. Topol Methods Nonlinear Anal 15:91–100

    MathSciNet  MATH  Google Scholar 

  • Pontryagin N (1968) Linear differential games, I–II. Soviet Math Dokl 8(3–4):769–771, 910–913

    Google Scholar 

  • Quincampoix M (1992) Differential inclusions and target problems. SIAM J Control Optim 30(2):324–335

    MathSciNet  MATH  Google Scholar 

  • Quincampoix M, Veliov V (2005) Optimal control of uncertain systems with incomplete information for the disturbance. SIAM J Control Optim 43(4):1373–1399

    MathSciNet  MATH  Google Scholar 

  • Rainer C (2007) On two different approaches to nonzero-sum stochastic differential games. Appl Math Optim 56:131–144

    MathSciNet  MATH  Google Scholar 

  • Roxin E (1969) The axiomatic approach in differential games. J Optim Theory Appl 3:153–163

    MathSciNet  MATH  Google Scholar 

  • Subbotin AI (1995) Generalized solutions of first-order PDEs. The dynamical optimization perspective. Translated from the Russian. Systems & control: foundations & applications. Birkhäuser, Boston

    Google Scholar 

  • Varaiya P (1967) The existence of solution to a differential game. SIAM J Control Optim 5:153–162

    MathSciNet  MATH  Google Scholar 

  • Varaiya P, Lin J (1967) Existence of saddle points i n differential game. SIAM J Control Optim 7(1):141–157

    MATH  Google Scholar 

  • Von Neumann J, Morgenstern O (1946) Theory of games and economic behaviour. Princeton University Press, Princeton

    Google Scholar 

Books and Reviews

  • Bardi M, Raghavan TES, Parthasarath T (eds) (1999) Stochastic and differential games. Theory and numerical methods. Dedicated to Prof. A. I. Subbotin. Annals of the International Society of Dynamic Games, vol 4. Birkhäuser, Boston

    Google Scholar 

  • Basar T, Olsder GJ (1999) Dynamic noncooperative game theory, 2nd edn. Classics in applied mathematics, vol 23. SIAM, Society for Industrial and Applied Mathematics, Philadelphia

    Google Scholar 

  • Blaquière A, Gérard F, Leitman G (1969) Quantitative and qualitative games. Academic, New York

    Google Scholar 

  • Buckdahn R, Cardaliaguet P, Quincampoix M (2011) Some recent aspects of differential game theory. Dyn Games Appl 1(1):74–114

    MathSciNet  MATH  Google Scholar 

  • Dockner E, Jörgensen S, Van Long N, Sorger G (2000) Differential games in economics and management science. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Hajek O (1975) Pursuit games. Academic, New York

    MATH  Google Scholar 

  • Isaacs R (1965) Differential games. Wiley, New York

    MATH  Google Scholar 

  • Jorgensen S, Quincampoix M, Vincent T (eds) (2007) Advances in dynamic games theory. Annals of International Society of Dynamic Games. Birkhäuser, Boston

    Google Scholar 

  • Krassovski NN, Subbotin AI (1988) Game-theorical control problems. Springer, New York

    Google Scholar 

  • Melikyan AA (1998) Generalized characteristics of first order PDEs. Applications in optimal control and differential games. Birkhäuser, Boston

    MATH  Google Scholar 

  • Patsko VS, Turova VL (2000) Numerical study of differential games with the homicidal chauffeur dynamics. Russian Academy of Sciences, Institute of Mathematics and Mechanics, Ekaterinburg

    MATH  Google Scholar 

  • Petrosyan LA (1993) Differential games of pursuit. Series on optimization, vol 2. World Scientific, Singapore

    Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), Université de Brest, Brest, France

    Marc Quincampoix

Authors
  1. Marc Quincampoix
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Corresponding author

Correspondence to Marc Quincampoix .

Editor information

Editors and Affiliations

  1. Department of Economics, University of Sao Paulo, EPGE Brazilian School of Economics and Finance, Sao Paulo, Brazil

    Marilda Sotomayor

  2. Dept. of Economics and Economic History, Universitat Autònoma de Barcelona and Barcelona GSE, Barcelona, Spain

    David Pérez-Castrillo

  3. Istituto Applicazioni del Calcolo (IAC), Consiglio Nazionale delle Ricerche (CNR), Rome, Italy

    Filippo Castiglione

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Quincampoix, M. (2020). Differential Games. In: Sotomayor, M., Pérez-Castrillo, D., Castiglione, F. (eds) Complex Social and Behavioral Systems . Encyclopedia of Complexity and Systems Science Series. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-0368-0_123

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