zbMATH — the first resource for mathematics

Some recent aspects of differential game theory. (English) Zbl 1214.91013
Summary: 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.

91A23 Differential games (aspects of game theory)
91A15 Stochastic games, stochastic differential games
60H30 Applications of stochastic analysis (to PDEs, etc.)
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
Full Text: DOI
[1] Achdou Y, Capuzzo Dolcetta I Mean field games: numerical methods. Preprint hal-00392074 · Zbl 1217.91019
[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 · Zbl 1032.35103
[4] Alvarez O, Bardi M (2007) Ergodic problems in differential games. In: Ann internat soc dynam games, vol 9. Birkhäuser, Boston, pp 131–152 · Zbl 1153.91346
[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 · Zbl 1209.35001
[6] Alziary de Roquefort B (1991) Jeux différentiels et approximation numérique de fonctions valeur. RAIRO Math Model Numer Anal 25:517–560 · Zbl 0734.90132
[7] Arisawa M, Lions P-L (1998) On ergodic stochastic control. Commun Partial Differ Equ 23(11–12):2187–2217 · Zbl 1126.93434
[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] 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
[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 · Zbl 0828.90142
[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 · Zbl 0853.90133
[15] Baras JS, Patel NS (1998) Robust control of set-valued discrete-time dynamical systems. IEEE Trans Automat Control 43:61–75 · Zbl 0907.93037
[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 · Zbl 1188.49030
[17] Bardi M, Capuzzo Dolcetta I (1996) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, Basel · Zbl 0890.49011
[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 · Zbl 1158.91323
[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 · Zbl 0878.60036
[21] Bensoussan A, Frehse J (2000) Stochastic games for N players. J Optim Theory Appl 105(3):543–565 · Zbl 0977.91006
[22] Bensoussan A, Frehse J (2002) Regularity results for nonlinear elliptic systems and applications. Applied mathematical sciences, vol 151. Springer, Berlin · Zbl 1055.35002
[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 · Zbl 1087.35014
[25] Bettiol P, Frankowska H (2007) Regularity of solution maps of differential inclusions under state constraints. Set-Valued Anal 15(1):21–45 · Zbl 1126.34011
[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 · Zbl 1103.49019
[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 · Zbl 0794.90086
[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 · Zbl 0387.90118
[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 · Zbl 1124.91011
[33] Bressan A, Shen W (2004) Small BV solutions of hyperbolic non-cooperative differential games. SIAM J Control Optim 43:104–215 · Zbl 1101.91009
[34] Bressan A, Shen W (2004) Semi-cooperative strategies for differential games. Int J Game Theory 32:561–593 · Zbl 1098.91019
[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 · Zbl 1157.93040
[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 · Zbl 1169.93026
[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 · Zbl 1258.93118
[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 · Zbl 1101.91010
[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 · Zbl 1183.60022
[40] Cardaliaguet P (1996) A differential game with two players and one target. SIAM J Control Optim 34(4):1441–1460 · Zbl 0853.90136
[41] Cardaliaguet P (2007) Differential games with asymmetric information. SIAM J Control Optim 46(3):816–838 · Zbl 1293.91020
[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 · Zbl 1152.91359
[43] Cardaliaguet P (2008) Representation formulas for differential games with asymmetric information. J Optim Theory Appl 138(1):1–16 · Zbl 1146.91010
[44] Cardaliaguet P (2009) A double obstacle problem arising in differential game theory. J Math Anal Appl 360(1):95–107 · Zbl 1173.49033
[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 \(\mathbb{R}\)2/\(\mathbb{Z}\)2. 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 · Zbl 1084.91008
[48] Cardaliaguet P, Quincampoix M (2008) Deterministic differential games under probability knowledge of initial condition. Int Game Theory Rev 10(1):1–16 · Zbl 1152.91358
[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 · Zbl 0982.91014
[50] Cardaliaguet P, Quincampoix M, Saint-Pierre P (2000) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615–1632 · Zbl 1140.91320
[51] Cardaliaguet P, Rainer C (2009) Stochastic differential games with asymmetric information. Appl Math Optim 59:1–36 · Zbl 1170.91308
[52] Cardaliaguet P, Rainer C (2009) On a continuous time game with incomplete information. Math Oper Res 34(4):769–794 · Zbl 1232.91008
[53] Cardaliaguet P, Souquière A (2010) Differential games with a blind player. Preprint · Zbl 1259.49062
[54] Case JH (1969) Toward a theory of many player differential games. SIAM J Control 7:179–197 · Zbl 0176.39404
[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 · Zbl 0755.35015
[57] De Meyer B (1996) Repeated games, duality and the central limit theorem. Math Oper Res 21(1):237–251 · Zbl 0846.90143
[58] De Meyer B, Rosenberg D (1999) ”Cav u” and the dual game. Math Oper Res 24(3):619–626 · Zbl 0966.91015
[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 · Zbl 0866.00052
[61] Evans LC (1989) The perturbed test function method for viscosity solutions of nonlinear PDE. Proc R Soc Edinb Sect A 111:359–375 · Zbl 0679.35001
[62] Evans LC (1992) Periodic homogenisation of certain fully nonlinear partial differential equations. Proc R Soc Edinb Sect A 120:245–265 · Zbl 0796.35011
[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 · Zbl 0686.90049
[66] Flynn J (1973) Lion and man: the boundary constraints. SIAM J Control 11:397 · Zbl 0259.90060
[67] Friedman A (1971) Differential games. Wiley, New York · Zbl 0229.90060
[68] Gaitsgory V, Nitzan S (1994) A folk theorem for dynamic games. J Math Econ 23(2):167–178 · Zbl 0802.90146
[69] Ghassemi KH (2005) Differential games of fixed duration with state constraints. J Oper Theory Appl 68(3):513–537 · Zbl 0699.90103
[70] Ghosh MK, Rao KSM (2005) Differential game with ergodic payoff. SIAM J Control Optim 43:2020–2035 · Zbl 1102.91020
[71] Gomes DA, Mohr J, Souza RR (2010) Discrete time, finite state space mean field games. J Math Pures Appl 93(3):308–328 · Zbl 1192.91028
[72] Guéant O (2009) Mean field games and applications to economics. PhD thesis, Université Paris-Dauphine · Zbl 1173.91020
[73] Guéant O (2009) A reference case for mean field games models. J Math Pures Appl (9) 92(3):276–294 · Zbl 1173.91020
[74] Hamadène S (1999) Nonzero sum linear-quadratic stochastic differential games and backward-forward equations. Stoch Anal Appl 17(1):117–130 · Zbl 0922.60050
[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 · Zbl 0892.60062
[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 {\(\epsilon\)}-Nash equilibria. IEEE Trans Autom Control 52(9):1560–1571 · Zbl 1366.91016
[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 · Zbl 1280.91020
[80] Isaacs R (1965) Differential games. Wiley, New York · Zbl 0144.12603
[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 · Zbl 0366.90141
[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 · Zbl 1193.91018
[86] Laraki R (2002) Repeated games with lack of information on one side: the dual differential approach. Math Oper Res 27(2):419–440 · Zbl 1082.91509
[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 · Zbl 1153.91010
[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 · Zbl 1153.91009
[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 · Zbl 1160.91016
[90] Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2(1):229–260 · Zbl 1156.91321
[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 · Zbl 1173.91310
[93] Lions PL (1982) Generalized solution of Hamilton–Jacobi equations. Pitman, London · Zbl 0497.35001
[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 · Zbl 0927.60004
[97] Mannucci P (2004/05) Nonzero-sum stochastic differential games with discontinuous feedback. SIAM J Control Optim 43(4):1222–1233 · Zbl 1077.91011
[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 · Zbl 0232.90066
[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 · Zbl 1005.91023
[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 · Zbl 1154.91324
[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 · Zbl 0977.49018
[104] Quincampoix M, Renault J On the existence of a limit value in some non expansive optimal control problems. Preprint hal-00377857 · Zbl 1234.49003
[105] Rainer C (2007) On two different approaches to nonzero-sum stochastic differential games. Appl Math Optim 56(1):131–144 · Zbl 1284.91051
[106] Ramasubramanian S (2007) A d-person differential game with state space constraints. Appl Math Optim 56(3):312–342 · Zbl 1133.91327
[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 Mat Mekh USSR 52(2):141–146 · Zbl 0691.90098
[109] Sorin S (2002) A first course on zero-sum repeated games. Mathématiques & applications, vol 37. Springer, Berlin · Zbl 1005.91019
[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 · Zbl 1211.91067
[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 · Zbl 1211.91067
[113] Starr AW, Ho YC (1969) Nonzero-sum differential games. J Optim Theory Appl 3:184–206 · Zbl 0169.12301
[114] Tolwinski B, Haurie A, Leitmann G (1986) Cooperative equilibria in differential games. J Math Anal Appl 119:182–202 · Zbl 0607.90097
[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 · Zbl 0943.93002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.