Some recent aspects of differential game theory

Dynamic Games and Applications

Volumn 1, Issue 1, 2011, Pages 74-114

  Buckdahn, R ^a   Cardaliaguet, P ^b   Quincampoix, M ^a  

^a UNIVERSITÉ DE BRETAGNE OCCIDENTALE   (France)

^b UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

Backward stochastic differential equations; Differential game; Incomplete information; Mean field games; State constraints; System of hamilton jacobi equations; Viscosity solution

Indexed keywords

EID: 84860688174     PISSN: 21530785     EISSN: 21530793     Source Type: Journal    
DOI: 10.1007/s13235-010-0005-0     Document Type: Article

References (116)

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
4. Alvarez O, Bardi M (2007) Ergodic problems in differential games. In: Ann internat soc dynam games, vol 9. Birkhäuser, Boston, pp 131-152
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
6. Alziary de Roquefort B (1991) Jeux différentiels et approximation numérique de fonctions valeur. RAIRO Math Model Numer Anal 25:517-560
7. Arisawa M, Lions P-L (1998) On ergodic stochastic control. Commun Partial Differ Equ 23(11-12):2187-2217
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
10. Aubin J-P, Frankowska H (1992) Set-valued analysis. Birkhaüser, Basel
11. Aumann RJ, Maschler MB (1995) Repeated games with incomplete information. MIT Press, Cambridge. With the collaboration of Richard E Stearns
12. ∞-optimal control and related minimax design problems. A dynamic game approach, 2nd edn. Systems & control: foundations & applications. Birkhäuser, Boston
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
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
15. Baras JS, Patel NS (1998) Robust control of set-valued discrete-time dynamical systems. IEEE Trans Automat Control 43:61-75
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
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
19. Barles G (1994) Solutions de viscosité des équations de Hamilton-Jacobi. Springer, Berlin
20. Barles G, Buckdahn R, Pardoux E (1997) Backward stochastic differential equations and integral-partial differential equations. Stoch Stoch Rep 60:57-83
21. Bensoussan A, Frehse J (2000) Stochastic games for N players. J Optim Theory Appl 105(3):543-565
22. Bensoussan A, Frehse J (2002) Regularity results for nonlinear elliptic systems and applications. Applied mathematical sciences, vol 151. Springer, Berlin
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
25. Bettiol P, Frankowska H (2007) Regularity of solution maps of differential inclusions under state constraints. Set-Valued Anal 15(1):21-45
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
27. Borkar VS, Ghosh MK (1992) Stochastic differential games: an occupation measure based approach. J Optim Theory Appl 73:359-385. Errata corrige, (1996) 88:251-252
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
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
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
33. Bressan A, Shen W (2004) Small BV solutions of hyperbolic non-cooperative differential games. SIAM J Control Optim 43:104-215
34. Bressan A, Shen W (2004) Semi-cooperative strategies for differential games. Int J Game Theory 32:561-593
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
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
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
40. Cardaliaguet P (1996) A differential game with two players and one target. SIAM J Control Optim 34(4):1441-1460
41. Cardaliaguet P (2007) Differential games with asymmetric information. SIAM J Control Optim 46(3):816-838
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
43. Cardaliaguet P (2008) Representation formulas for differential games with asymmetric information. J Optim Theory Appl 138(1):1-16
44. Cardaliaguet P (2009) A double obstacle problem arising in differential game theory. J Math Anal Appl 360(1):95-107
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
48. Cardaliaguet P, Quincampoix M (2008) Deterministic differential games under probability knowledge of initial condition. Int Game Theory Rev 10(1):1-16
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
50. Cardaliaguet P, Quincampoix M, Saint-Pierre P (2000) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615-1632
51. Cardaliaguet P, Rainer C (2009) Stochastic differential games with asymmetric information. Appl Math Optim 59:1-36
52. Cardaliaguet P, Rainer C (2009) On a continuous time game with incomplete information. Math Oper Res 34(4):769-794
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
55. Coulomb J-M, Gaitsgory V (2000) On a class of Nash equilibria with memory strategies for nonzerosum 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
57. De Meyer B (1996) Repeated games, duality and the central limit theorem. Math Oper Res 21(1):237-251
58. De Meyer B, Rosenberg D (1999) "Cav u" and the dual game. Math Oper Res 24(3):619-626
59. Dunford N, Schwartz JT (1957) Linear operators. Part I: general theory. Wiley-Interscience, New York
60. El Karoui N, Mazliak L (eds) (1997) Backward stochastic differential equations. CRC Press, New York
61. Evans LC (1989) The perturbed test function method for viscosity solutions of nonlinear PDE. Proc R Soc Edinb Sect A 111:359-375
62. Evans LC (1992) Periodic homogenisation of certain fully nonlinear partial differential equations. Proc R Soc Edinb Sect A 120:245-265
63. Evans LC, Souganidis PE (1984) Differential games and representation formulas for solutions of Hamilton-Jacobi equations. Indiana Univ Math J 282:487-502
64. Fleming WH, Soner HM (1993) Controlled Markov processes and viscosity solution. Springer, New- York
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
66. Flynn J (1973) Lion and man: the boundary constraints. SIAM J Control 11:397
67. Friedman A (1971) Differential games. Wiley, New York
68. Gaitsgory V, Nitzan S (1994) A folk theorem for dynamic games. J Math Econ 23(2):167-178
69. Ghassemi KH (2005) Differential games of fixed duration with state constraints. J Oper Theory Appl 68(3):513-537
70. Ghosh MK, Rao KSM (2005) Differential game with ergodic payoff. SIAM J Control Optim 43:2020-2035
71. Gomes DA, Mohr J, Souza RR (2010) Discrete time, finite state space mean field games. J Math Pures Appl 93(3):308-328
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
74. Hamadène S (1999) Nonzero sum linear-quadratic stochastic differential games and backward-forward equations. Stoch Anal Appl 17(1):117-130
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
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
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
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
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
80. Isaacs R (1965) Differential games. Wiley, New York
81. Kleimonov AF (1993) Nonantagonist differential games. Nauka, Ural'skoe Otdelenie, Ekaterinburg (in Russian)
82. Kononenko AF (1976) Equilibrium positional strategies in nonantagonistic differential games. Dokl Akad Nauk SSSR 231(2):285-288
83. Krasovskii NN, Subbotin AI (1988) Game-theorical control problems. Springer, New York
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
86. Laraki R (2002) Repeated games with lack of information on one side: the dual differential approach. Math Oper Res 27(2):419-440
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
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
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
90. Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2(1):229-260
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
93. Lions PL (1982) Generalized solution of Hamilton-Jacobi equations. Pitman, London
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
97. Mannucci P (2004/05) Nonzero-sum stochastic differential games with discontinuous feedback. SIAM J Control Optim 43(4):1222-1233
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
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
100. Osborne MJ (2004) An introduction to game theory. Oxford University Press, Oxford
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
102. Petrosjan LA (2004) Cooperation in games with incomplete information. In: Nonlinear analysis and convex analysis. Yokohama Publ, Yokohama, pp 469-479
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
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
106. Ramasubramanian S (2007) A d-person differential game with state space constraints. Appl Math Optim 56(3):312-342
107. Rapaport A, Bernhard P (1995) On a planar pursuit game with imperfect knowledge of a coordinate. Autom Prod Inform Ind 29:575-601
108. Rozyev I, Subbotin AI (1988) Semicontinuous solutions of Hamilton-Jacobi equations. Prikl MatMekh USSR 52(2):141-146
109. Sorin S (2002) A first course on zero-sum repeated games. Mathématiques & applications, vol 37. Springer, Berlin
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
114. Tolwinski B, Haurie A, Leitmann G (1986) Cooperative equilibria in differential games. J Math Anal Appl 119:182-202
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