Skip to main content
SpringerLink
Log in

Some Recent Aspects of Differential Game Theory

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

This survey paper presents some new advances in theoretical aspects of differential game theory. We particular focus on three topics: differential games with state constraints; backward stochastic differential equations approach to stochastic differential games; differential games with incomplete information. We also address some recent development in nonzero-sum differential games (analysis of systems of Hamilton–Jacobi equations by conservation laws methods; differential games with a large number of players, i.e., mean-field games) and long-time average of zero-sum differential games.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achdou Y, Capuzzo Dolcetta I Mean field games: numerical methods. Preprint hal-00392074

  2. Achdou Y, Camilli F, Capuzzo Dolcetta I (2010) Mean field games: numerical methods for the planning problem. Preprint

  3. Alvarez O, Bardi M (2003) Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch Ration Mech Anal 170(1):17–61

    Article  MATH  MathSciNet  Google Scholar 

  4. Alvarez O, Bardi M (2007) Ergodic problems in differential games. In: Ann internat soc dynam games, vol 9. Birkhäuser, Boston, pp 131–152

    Google Scholar 

  5. Alvarez O, Bardi M (2010) Ergodicity, stabilization, and singular perturbations for Bellman–Isaacs equations. Mem Am Math Soc, vol 204, no 960, vi+77 pp

    Google Scholar 

  6. Alziary de Roquefort B (1991) Jeux différentiels et approximation numérique de fonctions valeur. RAIRO Math Model Numer Anal 25:517–560

    MATH  MathSciNet  Google Scholar 

  7. Arisawa M, Lions P-L (1998) On ergodic stochastic control. Commun Partial Differ Equ 23(11–12):2187–2217

    MATH  MathSciNet  Google Scholar 

  8. As Soulaimani S (2008) Approchabilité, viabilité et jeux différentiels en information incomplète. Thèse de l’Université de Brest

  9. Aubin J-P (1992) Viability theory. Birkhaüser, Basel

    Google Scholar 

  10. Aubin J-P, Frankowska H (1992) Set-valued analysis. Birkhaüser, Basel

    Google Scholar 

  11. Aumann RJ, Maschler MB (1995) Repeated games with incomplete information. MIT Press, Cambridge. With the collaboration of Richard E Stearns

    MATH  Google Scholar 

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

    Google Scholar 

  13. Baçar T, Olsder GJ (1999) Dynamic noncooperative game theory. Classics in applied mathematics, vol 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. Reprint of the second (1995) edition

    Google Scholar 

  14. Baras JS, James MR (1996) Partially observed differential games, infinite-dimensional Hamilton–Jacobi–Isaacs equations, and nonlinear H ∞ control. SIAM J Control Optim 34:1342–1364

    Article  MATH  MathSciNet  Google Scholar 

  15. Baras JS, Patel NS (1998) Robust control of set-valued discrete-time dynamical systems. IEEE Trans Automat Control 43:61–75

    Article  MATH  MathSciNet  Google Scholar 

  16. Bardi M On differential games with long-time-average cost. Advances in dynamic games and their applications. In: Ann internat soc dynam games, vol 10. Birkhäuser, Boston, pp 3–18

  17. Bardi M, Capuzzo Dolcetta I (1996) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, Basel

    Google Scholar 

  18. Bardi M, Koike S, Soravia P (2000) Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations. Discrete Contin Dyn Syst 6(2):361–380

    Article  MATH  MathSciNet  Google Scholar 

  19. Barles G (1994) Solutions de viscosité des équations de Hamilton–Jacobi. Springer, Berlin

    MATH  Google Scholar 

  20. Barles G, Buckdahn R, Pardoux E (1997) Backward stochastic differential equations and integral-partial differential equations. Stoch Stoch Rep 60:57–83

    MATH  MathSciNet  Google Scholar 

  21. Bensoussan A, Frehse J (2000) Stochastic games for N players. J Optim Theory Appl 105(3):543–565

    Article  MATH  MathSciNet  Google Scholar 

  22. Bensoussan A, Frehse J (2002) Regularity results for nonlinear elliptic systems and applications. Applied mathematical sciences, vol 151. Springer, Berlin

    MATH  Google Scholar 

  23. Bernhard P (1979) Contribution à l’étude des jeux différentiels à some nulle et information parfaite. Thèse Université de Paris VI

  24. Bettiol P (2005) On ergodic problem for Hamilton–Jacobi–Isaacs equations. ESAIM Control Optim Calc Var 11:522–541

    Article  MATH  MathSciNet  Google Scholar 

  25. Bettiol P, Frankowska H (2007) Regularity of solution maps of differential inclusions under state constraints. Set-Valued Anal 15(1):21–45

    Article  MATH  MathSciNet  Google Scholar 

  26. Bettiol P, Cardaliaguet P, Quincampoix M (2006) Zero-sum state constraint differential game: existence of a value for Bolza problem. Int J Game Theory 34(3):495–527

    Article  MATH  MathSciNet  Google Scholar 

  27. Borkar VS, Ghosh MK (1992) Stochastic differential games: an occupation measure based approach. J Optim Theory Appl 73:359–385. Errata corrige, ibid (1996) 88:251–252

    Article  MATH  MathSciNet  Google Scholar 

  28. 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 

  29. Breakwell JV (1989) Time-optimal pursuit inside a circle. In: Differential games and applications, Sophia-Antipolis, 1988. Lecture notes in control and inform sci, vol 119. Springer, Berlin, pp 72–85

    Chapter  Google Scholar 

  30. Bressan A From optimal control to non-cooperative differential games: a homotopy approach. Control Cybern, to appear

  31. Bressan A Bifurcation analysis of a noncooperative differential game with one weak player. J Differ Equ. doi:10.1016/j.jde.2009.11.025

  32. Bressan A, Priuli F (2006) Infinite horizon noncooperative differential games. J Differ Equ 227:230–257

    Article  MATH  MathSciNet  Google Scholar 

  33. Bressan A, Shen W (2004) Small BV solutions of hyperbolic non-cooperative differential games. SIAM J Control Optim 43:104–215

    Article  MathSciNet  Google Scholar 

  34. Bressan A, Shen W (2004) Semi-cooperative strategies for differential games. Int J Game Theory 32:561–593

    Article  MATH  MathSciNet  Google Scholar 

  35. Buckdahn R, Li J (2008) Stochastic differential games and viscosity solutions for Hamilton–Jacobi–Bellman–Isaacs equations. SIAM J Control Optim 47(1):444–475

    Article  MATH  MathSciNet  Google Scholar 

  36. Buckdahn R, Li J Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers. Nonlinear Differ Equ Appl. doi:10.1007/s00030-009-0022-0

  37. Buckdahn R, Li J Stochastic differential games with reflection and related obstacle problems for Isaacs equations. Acta Math Appl Sin, Engl Ser, accepted for publication

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

    Article  MATH  MathSciNet  Google Scholar 

  39. Buckdahn R, Li J, Peng S (2009) Mean-field backward stochastic differential equations and related partial differential equations. Stoch Process Appl 119(10):3133–3154

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  42. Cardaliaguet P (2007) On the instability of the feedback equilibrium payoff in a nonzero-sum differential game on the line. In: Advances in dynamic game theory. Ann internat soc dynam games, vol 9. Birkhäuser, Boston, pp 57–67

    Chapter  Google Scholar 

  43. Cardaliaguet P (2008) Representation formulas for differential games with asymmetric information. J Optim Theory Appl 138(1):1–16

    Article  MATH  MathSciNet  Google Scholar 

  44. Cardaliaguet P (2009) A double obstacle problem arising in differential game theory. J Math Anal Appl 360(1):95–107

    Article  MATH  MathSciNet  Google Scholar 

  45. Cardaliaguet P Numerical approximation and optimal strategies for differential games with lack of information on one side. Ann ISDG. doi:10.1007/978-0-8176-4834-3_10

  46. Cardaliaguet P Ergodicity of Hamilton–Jacobi equations with a non coercive non convex Hamiltonian in ℝ2/ℤ2. Ann Inst H Poincaré. doi:10.1016/j.anihpc.2009.11.015

  47. Cardaliaguet P, Plaskacz S (2003) Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game. Int J Game Theory 32:33–71

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  49. Cardaliaguet P, Quincampoix M, Saint-Pierre P (1999) Numerical methods for differential games. In: Stochastic and differential games: Theory and numerical methods. Annals of the international society of dynamic games. Birkhäuser, Basel, pp 177–247

    Google Scholar 

  50. Cardaliaguet P, Quincampoix M, Saint-Pierre P (2000) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615–1632

    Article  MathSciNet  Google Scholar 

  51. Cardaliaguet P, Rainer C (2009) Stochastic differential games with asymmetric information. Appl Math Optim 59:1–36

    Article  MATH  MathSciNet  Google Scholar 

  52. Cardaliaguet P, Rainer C (2009) On a continuous time game with incomplete information. Math Oper Res 34(4):769–794

    Article  MathSciNet  Google Scholar 

  53. Cardaliaguet P, Souquière A (2010) Differential games with a blind player. Preprint

  54. Case JH (1969) Toward a theory of many player differential games. SIAM J Control 7:179–197

    Article  MATH  MathSciNet  Google Scholar 

  55. Coulomb J-M, Gaitsgory V (2000) On a class of Nash equilibria with memory strategies for nonzero-sum differential games In: Recent research in dynamic games (Ischia, 1999). Int Game Theory Rev 2(2–3):173–192

  56. Crandall MG, Ishii H, Lions P-L (1992) User’s guide to viscosity solutions of second order partial differential equations. Bull Am Soc 27:1–67

    Article  MATH  MathSciNet  Google Scholar 

  57. De Meyer B (1996) Repeated games, duality and the central limit theorem. Math Oper Res 21(1):237–251

    Article  MATH  MathSciNet  Google Scholar 

  58. De Meyer B, Rosenberg D (1999) “Cav u” and the dual game. Math Oper Res 24(3):619–626

    Article  MATH  MathSciNet  Google Scholar 

  59. Dunford N, Schwartz JT (1957) Linear operators. Part I: general theory. Wiley-Interscience, New York

    Google Scholar 

  60. El Karoui N, Mazliak L (eds) (1997) Backward stochastic differential equations. CRC Press, New York

    MATH  Google Scholar 

  61. Evans LC (1989) The perturbed test function method for viscosity solutions of nonlinear PDE. Proc R Soc Edinb Sect A 111:359–375

    MATH  Google Scholar 

  62. Evans LC (1992) Periodic homogenisation of certain fully nonlinear partial differential equations. Proc R Soc Edinb Sect A 120:245–265

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  64. Fleming WH, Soner HM (1993) Controlled Markov processes and viscosity solution. Springer, New-York

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  67. Friedman A (1971) Differential games. Wiley, New York

    MATH  Google Scholar 

  68. Gaitsgory V, Nitzan S (1994) A folk theorem for dynamic games. J Math Econ 23(2):167–178

    MATH  MathSciNet  Google Scholar 

  69. Ghassemi KH (2005) Differential games of fixed duration with state constraints. J Oper Theory Appl 68(3):513–537

    Article  MathSciNet  Google Scholar 

  70. Ghosh MK, Rao KSM (2005) Differential game with ergodic payoff. SIAM J Control Optim 43:2020–2035

    Article  MATH  MathSciNet  Google Scholar 

  71. Gomes DA, Mohr J, Souza RR (2010) Discrete time, finite state space mean field games. J Math Pures Appl 93(3):308–328

    Article  MATH  MathSciNet  Google Scholar 

  72. Guéant O (2009) Mean field games and applications to economics. PhD thesis, Université Paris-Dauphine

  73. Guéant O (2009) A reference case for mean field games models. J Math Pures Appl (9) 92(3):276–294

    Article  MATH  MathSciNet  Google Scholar 

  74. Hamadène S (1999) Nonzero sum linear-quadratic stochastic differential games and backward-forward equations. Stoch Anal Appl 17(1):117–130

    Article  MATH  Google Scholar 

  75. Hamadène S, Lepeltier J-P, Peng S (1997) BSDEs with continuous coefficients and stochastic differential games. In: El Karoui N et al. (eds) Backward stochastic differential equations. Pitman res notes math ser, vol 364. Longman, Harlow, pp 115–128

    Google Scholar 

  76. Huang M, Caines PE, Malhamé RP (2003) Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. In: Proc 42nd IEEE conf decision contr, Maui, Hawaii, Dec 2003, pp 98–103

    Google Scholar 

  77. Huang M, Caines PE, Malhame RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ε-Nash equilibria. IEEE Trans Autom Control 52(9):1560–1571

    Article  MathSciNet  Google Scholar 

  78. Huang M, Caines PE, Malhame RP (2007) The Nash certainty equivalence principle and McKean–Vlasov systems: an invariance principle and entry adaptation. In: 46th IEEE conference on decision and control, pp 121–123

    Chapter  Google Scholar 

  79. Huang M, Caines PE, Malhame RP (2007) An invariance principle in large population stochastic dynamic games. J Syst Sci Complex 20(2):162–172

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  81. Kleimonov AF (1993) Nonantagonist differential games. Nauka, Ural’skoe Otdelenie, Ekaterinburg (in Russian)

    Google Scholar 

  82. Kononenko AF (1976) Equilibrium positional strategies in nonantagonistic differential games. Dokl Akad Nauk SSSR 231(2):285–288

    MathSciNet  Google Scholar 

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

    Google Scholar 

  84. Lachapelle A Human crowds and groups interactions: a mean field games approach. Preprint

  85. Lachapelle A, Salomon J, Turinici G (2010) Computation of mean field equilibria in economics. Math Models Methods Appl Sci 1:1–22

    MathSciNet  Google Scholar 

  86. Laraki R (2002) Repeated games with lack of information on one side: the dual differential approach. Math Oper Res 27(2):419–440

    Article  MATH  MathSciNet  Google Scholar 

  87. Lasry J-M, Lions P-L (2006) Jeux a champ moyen. II. Horizon fini et contrôle optimal. C R Math Acad Sci Paris 343(10):679–684

    MATH  MathSciNet  Google Scholar 

  88. Lasry J-M, Lions P-L (2006) Jeux a champ moyen. I. Le cas stationnaire. C R Math Acad Sci Paris 343(9):619–625

    MATH  MathSciNet  Google Scholar 

  89. Lasry J-M, Lions P-L (2007) Large investor trading impacts on volatility. Ann Inst H Poincaré Anal Non Linéaire 24(2):311–323

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  91. Lasry J-M, Lions P-L, Guéant O (2009) Application of mean field games to growth theory. Preprint

  92. Lepeltier J-P, Wu Z, Yu Z (2009) Nash equilibrium point for one kind of stochastic nonzero-sum game problem and BSDEs. C R Math Acad Sci Paris 347(15–16):959–964

    MATH  MathSciNet  Google Scholar 

  93. Lions PL (1982) Generalized solution of Hamilton–Jacobi equations. Pitman, London

    Google Scholar 

  94. Lions P-L (2007–2010) Cours au Collège de France

  95. Lions P-L, Papanicolaou G, Varadhan SRS Homogenization of Hamilton–Jacobi equations. Unpublished work

  96. Ma J, Yong J (2007) Forward–backward stochastic differential equations and their applications. Springer, Berlin

    Book  Google Scholar 

  97. Mannucci P (2004/05) Nonzero-sum stochastic differential games with discontinuous feedback. SIAM J Control Optim 43(4):1222–1233

    Article  MathSciNet  Google Scholar 

  98. Mertens JF, Zamir S (1994) The value of two person zero sum repeated games with lack of information on both sides. Int J Game Theory 1:39–64

    Article  MathSciNet  Google Scholar 

  99. Olsder GJ (2001/2002) On open and closed loop bang-bang control in nonzero-sum differential games. SIAM J Control Optim 40:1087–1106

    Article  MathSciNet  Google Scholar 

  100. Osborne MJ (2004) An introduction to game theory. Oxford University Press, Oxford

    Google Scholar 

  101. Peng S (1997) BSDE and stochastic optimizations. In: Yan J, Peng S, Fang S, Wu L (eds) Topics in stochastic analysis. Science Press, Beijing

    Google Scholar 

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

    Google Scholar 

  103. Plaskacz S, Quincampoix M (2000) Value-functions for differential games and control systems with discontinuous terminal cost. SIAM J Control Optim 39(5):1485–1498

    Article  MathSciNet  Google Scholar 

  104. Quincampoix M, Renault J On the existence of a limit value in some non expansive optimal control problems. Preprint hal-00377857

  105. Rainer C (2007) On two different approaches to nonzero-sum stochastic differential games. Appl Math Optim 56(1):131–144

    Article  MATH  MathSciNet  Google Scholar 

  106. Ramasubramanian S (2007) A d-person differential game with state space constraints. Appl Math Optim 56(3):312–342

    Article  MATH  MathSciNet  Google Scholar 

  107. Rapaport A, Bernhard P (1995) On a planar pursuit game with imperfect knowledge of a coordinate. Autom Prod Inform Ind 29:575–601

    Google Scholar 

  108. Rozyev I, Subbotin AI (1988) Semicontinuous solutions of Hamilton–Jacobi equations. Prikl Mat Mekh USSR 52(2):141–146

    MATH  MathSciNet  Google Scholar 

  109. Sorin S (2002) A first course on zero-sum repeated games. Mathématiques & applications, vol 37. Springer, Berlin

    MATH  Google Scholar 

  110. Souquière A (2008) Approximation and representation of the value for some differential games with imperfect information. Int J Game Theory. doi:10.1007/s00182-009-0217-y

  111. Souquière A (2010) Nash and publicly correlated equilibrium payoffs in non-zero sum differential games using mixed strategies. Preprint

  112. Souquière A (2010) Jeux différentiels a information imparfaite. PhD Thesis, Brest University

  113. Starr AW, Ho YC (1969) Nonzero-sum differential games. J Optim Theory Appl 3:184–206

    Article  MATH  MathSciNet  Google Scholar 

  114. Tolwinski B, Haurie A, Leitmann G (1986) Cooperative equilibria in differential games. J Math Anal Appl 119:182–202

    Article  MATH  MathSciNet  Google Scholar 

  115. Yin H, Mehta PG, Meyn SP, Shanbhag UV Synchronization of coupled oscillators is a game

  116. Yong J, Zhou XY (1999) Stochastic controls: Hamiltonian systems and HJB equations. Springer, Berlin

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques, UMR 6205, Université de Brest, 6 Av. Le Gorgeu, BP 809, 29285, Brest, France

    R. Buckdahn & M. Quincampoix

  2. Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    P. Cardaliaguet

Authors
  1. R. Buckdahn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Cardaliaguet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Quincampoix
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Cardaliaguet.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Buckdahn, R., Cardaliaguet, P. & Quincampoix, M. Some Recent Aspects of Differential Game Theory. Dyn Games Appl 1, 74–114 (2011). https://doi.org/10.1007/s13235-010-0005-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-010-0005-0

Keywords

Search

Navigation