Limit Value of Dynamic Zero-Sum Games with Vanishing Stage Duration

Published Online:https://doi.org/10.1287/moor.2017.0851

References

  • As Soulaimani S (2008) Viability with probabilistic knowledge of initial condition, application to optimal control. Set-Valued Anal. 16(7–8):1037–1060.CrossrefGoogle Scholar
  • Bardi M, Capuzzo-Dolcetta I (1996) Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Birkhauser, Boston).Google Scholar
  • Barron EN, Evans LC, Jensen R (1984) Viscosity solutions of Isaacs’ equations and differential games with Lipschitz controls. J. Differential Equations 53(2):213–233.CrossrefGoogle Scholar
  • Buckdahn R, Li J, Quincampoix M (2013) Value function of differential games without Isaacs’ conditions. An approach with nonanticipative mixed strategies. Internat. J. Game Theory 42(4):989–1020.CrossrefGoogle Scholar
  • Buckdahn R, Quincampoix M, Rainer C, Xu Y (2016) Differential games with asymmetric information and without Isaacs’ condition. Internat. J. Game Theory 45(4):795–816.CrossrefGoogle Scholar
  • Cardaliaguet P (2010) Introduction to Differential Games. Unpublished lecture notes.Google Scholar
  • Cardaliaguet P, Quincampoix M (2008) Deterministic differential games under probability knowledge of initial condition. Internat. Game Theory Rev. 10(1):1–16.CrossrefGoogle Scholar
  • Cardaliaguet P, Laraki R, Sorin S (2012) A continuous time approach for the asymptotic value in two-person zero-sum repeated games. SIAM J. Control Optim. 50(3):1573–1596.CrossrefGoogle Scholar
  • Cardaliaguet P, Rainer C, Rosenberg D, Vieille N (2014) Markov games with frequent actions and incomplete information: The limit case. Math. Oper. Res. 41(1):9–71.Google Scholar
  • Crandall MG, Lions P-L (1981) Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre. C.R. Acad. Sci. Paris 292:183–186.Google Scholar
  • Crandall MG, Ishii H, Lions P-L (1992) User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27(1):1–67.CrossrefGoogle Scholar
  • Elliot RJ, Kalton NJ (1972) The Existence of Value in Differential Games. Memoirs of the American Mathematical Society, No. 126 (AMS, Providence, RI).Google Scholar
  • Evans LC, Souganidis PE (1984) Differential games and representation formulas for solutions of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33(5):773–797.CrossrefGoogle Scholar
  • Fleming WH (1957) A note on differential games of prescribed duration. Dresher M, Tucker AW, Wolfe P, eds. Contributions to the Theory of Games, III, Annals of Mathematical Studies, Vol. 39 (Princeton University Press, Princeton, NJ), 407–412.Google Scholar
  • Fleming WH (1961) The convergence problem for differential games. J. Math. Anal. Appl. 3(1):102–116.CrossrefGoogle Scholar
  • Fleming WH (1964) The convergence problem for differential games II. Dresher M, Shapley LS, Tucker AW, eds. Advances in Game Theory, Annals of Mathematical Studies, Vol. 52 (Princeton University Press, Princeton, NJ), 195–210.CrossrefGoogle Scholar
  • Friedman A (1971) Differential Games (John Wiley & Sons, New York).Google Scholar
  • Friedman A (1974) Differential Games, CBMS Regional Conference Series in Mathematics, Vol. 18 (AMS, Providence, RI).Google Scholar
  • Guo X, Hernandez-Lerma O (2003) Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates. J. Appl. Probab. 40(2):327–345.CrossrefGoogle Scholar
  • Guo X, Hernandez-Lerma O (2005) Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates. Bernoulli 11(6):1009–1029.CrossrefGoogle Scholar
  • Isaacs R (1965) Differential Games (John Wiley & Sons, New York).Google Scholar
  • Jimenez C, Quincampoix M, Xu Y (2016) Differential games with incomplete information on a continuum of initial positions and without Isaacs condition. Dynam. Games Appl. 6(1):82–96.CrossrefGoogle Scholar
  • Laraki R (2002) Repeated games with lack of information on one side: The dual differential approach. Math. Oper. Res. 27(2):419–440.LinkGoogle Scholar
  • Mertens J-F, Sorin S, Zamir S (2015) Repeated Games (Cambridge University Press, Cambridge, UK).CrossrefGoogle Scholar
  • Neyman A (2017) Continuous-time stochastic games. Games Econom. Behavior 104:92–130.CrossrefGoogle Scholar
  • Neyman A (2013) Stochastic games with short-stage duration. Dynam. Games Appl. 3(2):236–278.CrossrefGoogle Scholar
  • Neyman A, Sorin S, eds. (2003) Stochastic Games and Applications, NATO Science Series, Vol. C 570 (Kluwer Academic Publishers, Dordrecht, Netherlands).CrossrefGoogle Scholar
  • Prieto-Rumeau T, Hernandez-Lerma O (2012) Selected Topics on Continuous-Time Controlled Markov Chains and Markov Games (Imperial College Press, London).CrossrefGoogle Scholar
  • Rosenberg D, Sorin S (2001) An operator approach to zero-sum repeated games. Israël J. Math. 121(1):221–246.CrossrefGoogle Scholar
  • Scarf HE (1957) On differential games with survival payoff. Dresher M, Tucker AW, Wolfe P, eds. Contributions to the Theory of Games, III, Annals of Mathematical Studies, Vol. 39 (Princeton University Press, Princeton, NJ), 393–405.Google Scholar
  • Shapley LS (1953) Stochastic games. Proc. Natl. Acad. Sci. USA 39(10):1095–1100.CrossrefGoogle Scholar
  • Sorin S (1984) “Big Match” with lack of information on one side (Part I). Internat. J. Game Theory 13(4):201–255.CrossrefGoogle Scholar
  • Sorin S (2002) A First Course on Zero-Sum Repeated Games (Springer, Berlin).Google Scholar
  • Sorin S (2011) Zero-sum repeated games: Recent advances and new links with differential games. Dynam. Games Appl. 1(1):172–207.CrossrefGoogle Scholar
  • Sorin S (2017) Limit value of dynamic games zero-sum games with vanishing stage duration: The symmetric case. In preparation.Google Scholar
  • Sorin S, Vigeral G (2015) Reversibility and oscillations in zero-sum discounted stochastic games. J. Dynam. Games 2(1):103–115.CrossrefGoogle Scholar
  • Sorin S, Vigeral G (2016) Operator approach to values of stochastic games with varying stage duration. Internat. J. Game Theory 45(1):389–410.CrossrefGoogle Scholar
  • Souganidis PE (1985) Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differential Equations 59(1):1–43.CrossrefGoogle Scholar
  • Tanaka K, Wakuta K (1977) On continuous Markov games with the expected average reward criterion. Sci. Rep. Niigata Univ. Ser. A 14:15–24.Google Scholar
  • Vieille N (1992) Weak approachability. Math. Oper. Res. 17(4):781–791.LinkGoogle Scholar
  • Vigeral G (2013) A zero-sum stochastic game with compact action sets and no asymptotic value. Dynam. Games Appl. 3(2):172–186.CrossrefGoogle Scholar
  • Zachrisson LE (1964) Markov games. Dresher M, Shapley LS, Tucker AW, eds. Advances in Game Theory, Annals of Mathematical Studies, Vol. 52 (Princeton University Press, Princeton, NJ), 211–253.Google Scholar
  • Ziliotto B (2016) Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture maxmin = lim υn. Ann. Probab. 44(2):1107–1133.CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.