January  2014, 1(1): 153-179. doi: 10.3934/jdg.2014.1.153

Structure of approximate solutions of dynamic continuous time zero-sum games

1. 

Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel

Received  April 2012 Revised  June 2012 Published  June 2013

In this paper we study a turnpike property of approximate solutions for a class of dynamic continuous-time two-player zero-sum games. These properties describe the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.
Citation: Alexander J. Zaslavski. Structure of approximate solutions of dynamic continuous time zero-sum games. Journal of Dynamics and Games, 2014, 1 (1) : 153-179. doi: 10.3934/jdg.2014.1.153
References:
[1]

O. Alvarez and M. Bardi, Ergodic problems in differential games, in "Advances in Dynamic Game Theory," Ann. Internat. Soc. Dynam. Games, 9, Birkhäuser Boston, Boston, MA, (2007), 131-152. doi: 10.1007/978-0-8176-4553-3_7.

[2]

B. D. O. Anderson and J. B. Moore, "Linear Optimal Control," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1971.

[3]

J.-P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.

[4]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Physica D, 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.

[5]

M. Bardi, On differential games with long-time-average cost, in "Advances in Dynamic Games and their Applications," Ann. Internat. Soc. Dynam. Games, 10, Birkhäuser Boston, Inc., Boston, MA, (2009), 3-18.

[6]

J. Baumeister, A. Leitäo and G. N. Silva, On the value function for nonautonomous optimal control problem with infinite horizon, Systems Control Lett., 56 (2007), 188-196. doi: 10.1016/j.sysconle.2006.08.011.

[7]

J. Blot and P. Cartigny, Optimality in infinite-horizon variational problems under sign conditions, J. Optim. Theory Appl., 106 (2000), 411-419. doi: 10.1023/A:1004611816252.

[8]

J. Blot and N. Hayek, Sufficient conditions for infinite-horizon calculus of variations problems, ESAIM Control Optim. Calc. Var., 5 (2000), 279-292. doi: 10.1051/cocv:2000111.

[9]

P. Cartigny and P. Michel, On a sufficient transversality condition for infinite horizon optimal control problems, Automatica J. IFAC, 39 (2003), 1007-1010. doi: 10.1016/S0005-1098(03)00060-8.

[10]

L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations," Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983.

[11]

I. V. Evstigneev and S. D. Flåm, Rapid growth paths in multivalued dynamical systems generated by homogeneous convex stochastic operators, Set-Valued Anal., 6 (1998), 61-81. doi: 10.1023/A:1008606332037.

[12]

D. Gale, On optimal development in a multisector economy, Rev. of Econ. Studies, 34 (1967), 1-19.

[13]

M. K. Ghosh and K. S. Mallikarjuna Rao, Differential games with ergodic payoff, SIAM J. Control Optim., 43 (2005), 2020-2035. doi: 10.1137/S0363012903404511.

[14]

X. Guo and O. Hernández-Lerma, Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates, Bernoulli, 11 (2005), 1009-1029. doi: 10.3150/bj/1137421638.

[15]

H. Jasso-Fuentes and O. Hernández-Lerma, Characterizations of overtaking optimality for controlled diffusion processes, Appl. Math. Optim., 57 (2008), 349-369. doi: 10.1007/s00245-007-9025-6.

[16]

O. Hernández-Lerma and J. B. Lasserre, Zero-sum stochastic games in Borel spaces: Average payoff criteria, SIAM J. Control Optim., 39 (2000), 1520-1539. doi: 10.1137/S0363012999361962.

[17]

V. Kolokoltsov and W. Yang, The turnpike theorems for Markov games, Dynamic Games and Applications, 2 (2012), 294-312. doi: 10.1007/s13235-012-0047-6.

[18]

A. Leizarowitz, Infinite horizon autonomous systems with unbounded cost, Appl. Math. and Opt., 13 (1985), 19-43. doi: 10.1007/BF01442197.

[19]

A. Leizarowitz and V. J. Mizel, One dimensional infinite horizon variational problems arising in continuum mechanics, Arch. Rational Mech. Anal., 106 (1989), 161-194. doi: 10.1007/BF00251430.

[20]

V. Lykina, S. Pickenhain and M. Wagner, Different interpretations of the improper integral objective in an infinite horizon control problem, J. Math. Anal. Appl., 340 (2008), 498-510. doi: 10.1016/j.jmaa.2007.08.008.

[21]

V. L. Makarov and A. M. Rubinov, "Mathematical Theory of Economic Dynamics and Equilibria," Springer-Verlag, New York-Heidelberg, 1977.

[22]

M. Marcus and A. J. Zaslavski, The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 16 (1999), 593-629. doi: 10.1016/S0294-1449(99)80029-8.

[23]

L. W. McKenzie, Turnpike theory, Econometrica, 44 (1976), 841-865. doi: 10.2307/1911532.

[24]

L. W. McKenzie, "Classical General Equilibrium Theory," MIT press, Cambridge, MA, 2002.

[25]

B. Sh. Mordukhovich, Minimax sythesis of a class of control systems with distributed parameters, Automat. Remote Control, 50 (1989), 1333-1340.

[26]

B. S. Mordukhovich and I. Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations, in "Optimal Control, Stabilization and Nonsmooth Analysis," Lecture Notes Control Inform. Sci., 301, Springer, Berlin, (2004), 121-132. doi: 10.1007/978-3-540-39983-4_8.

[27]

S. Pickenhain, V. Lykina and M. Wagner, On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems, Control Cybernet, 37 (2008), 451-468.

[28]

T. Prieto-Rumeau and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games, Math. Methods Oper. Res., 61 (2005), 437-454. doi: 10.1007/s001860400392.

[29]

P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule, American Economic Review, 55 (1965), 486-496.

[30]

C. C. von Weizsacker, Existence of optimal programs of accumulation for an infinite horizon, Rev. Econ. Studies, 32 (1965), 85-104.

[31]

A. J. Zaslavski, Ground states in a model of Frenkel-Kontorova type, Math. USSR-Izvestiya, 29 (1987), 323-354. doi: 10.1070/IM1987v029n02ABEH000972.

[32]

A. J. Zaslavski, Optimal programs on infinite horizon. I, II, SIAM Journal on Control and Optimization, 3 (1995), 1643-1660, 1661-1686. doi: 10.1137/S036301299325726X.

[33]

A. J. Zaslavski, Dynamic properties of optimal solutions of variational problems, Nonlinear Analysis, 27 (1996), 895-931. doi: 10.1016/0362-546X(95)00029-U.

[34]

A. J. Zaslavski, Turnpike property for dynamic discrete time zero-sum games, Abstract and Applied Analysis, 4 (1999), 21-48. doi: 10.1155/S1085337599000020.

[35]

A. J. Zaslavski, "Turnpike Properties in the Calculus of Variations and Optimal Control," Nonconvex Optimization and its Applications, 80, Springer, New York, 2006 .

[36]

A. J. Zaslavski, A turnpike result for a class of problems of the calculus of variations with extended-valued integrands, J. Convex Analysis, 15 (2008), 869-890.

[37]

A. J. Zaslavski, "Optimization on Metric and Normed Spaces," Springer Optimization and Its Applications, 44, Springer, New York, 2010. doi: 10.1007/978-0-387-88621-3.

[38]

A. J. Zaslavski, The existence and structure of approximate solutions of dynamic discrete time zero-sum games, Journal of Nonlinear and Convex Analysis, 12 (2011), 49-68.

[39]

A. J. Zaslavski and A. Leizarowitz, Optimal solutions of linear control systems with nonperiodic integrands, Mathematics of Operations Research, 22 (1997), 726-746. doi: 10.1287/moor.22.3.726.

show all references

References:
[1]

O. Alvarez and M. Bardi, Ergodic problems in differential games, in "Advances in Dynamic Game Theory," Ann. Internat. Soc. Dynam. Games, 9, Birkhäuser Boston, Boston, MA, (2007), 131-152. doi: 10.1007/978-0-8176-4553-3_7.

[2]

B. D. O. Anderson and J. B. Moore, "Linear Optimal Control," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1971.

[3]

J.-P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.

[4]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Physica D, 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.

[5]

M. Bardi, On differential games with long-time-average cost, in "Advances in Dynamic Games and their Applications," Ann. Internat. Soc. Dynam. Games, 10, Birkhäuser Boston, Inc., Boston, MA, (2009), 3-18.

[6]

J. Baumeister, A. Leitäo and G. N. Silva, On the value function for nonautonomous optimal control problem with infinite horizon, Systems Control Lett., 56 (2007), 188-196. doi: 10.1016/j.sysconle.2006.08.011.

[7]

J. Blot and P. Cartigny, Optimality in infinite-horizon variational problems under sign conditions, J. Optim. Theory Appl., 106 (2000), 411-419. doi: 10.1023/A:1004611816252.

[8]

J. Blot and N. Hayek, Sufficient conditions for infinite-horizon calculus of variations problems, ESAIM Control Optim. Calc. Var., 5 (2000), 279-292. doi: 10.1051/cocv:2000111.

[9]

P. Cartigny and P. Michel, On a sufficient transversality condition for infinite horizon optimal control problems, Automatica J. IFAC, 39 (2003), 1007-1010. doi: 10.1016/S0005-1098(03)00060-8.

[10]

L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations," Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983.

[11]

I. V. Evstigneev and S. D. Flåm, Rapid growth paths in multivalued dynamical systems generated by homogeneous convex stochastic operators, Set-Valued Anal., 6 (1998), 61-81. doi: 10.1023/A:1008606332037.

[12]

D. Gale, On optimal development in a multisector economy, Rev. of Econ. Studies, 34 (1967), 1-19.

[13]

M. K. Ghosh and K. S. Mallikarjuna Rao, Differential games with ergodic payoff, SIAM J. Control Optim., 43 (2005), 2020-2035. doi: 10.1137/S0363012903404511.

[14]

X. Guo and O. Hernández-Lerma, Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates, Bernoulli, 11 (2005), 1009-1029. doi: 10.3150/bj/1137421638.

[15]

H. Jasso-Fuentes and O. Hernández-Lerma, Characterizations of overtaking optimality for controlled diffusion processes, Appl. Math. Optim., 57 (2008), 349-369. doi: 10.1007/s00245-007-9025-6.

[16]

O. Hernández-Lerma and J. B. Lasserre, Zero-sum stochastic games in Borel spaces: Average payoff criteria, SIAM J. Control Optim., 39 (2000), 1520-1539. doi: 10.1137/S0363012999361962.

[17]

V. Kolokoltsov and W. Yang, The turnpike theorems for Markov games, Dynamic Games and Applications, 2 (2012), 294-312. doi: 10.1007/s13235-012-0047-6.

[18]

A. Leizarowitz, Infinite horizon autonomous systems with unbounded cost, Appl. Math. and Opt., 13 (1985), 19-43. doi: 10.1007/BF01442197.

[19]

A. Leizarowitz and V. J. Mizel, One dimensional infinite horizon variational problems arising in continuum mechanics, Arch. Rational Mech. Anal., 106 (1989), 161-194. doi: 10.1007/BF00251430.

[20]

V. Lykina, S. Pickenhain and M. Wagner, Different interpretations of the improper integral objective in an infinite horizon control problem, J. Math. Anal. Appl., 340 (2008), 498-510. doi: 10.1016/j.jmaa.2007.08.008.

[21]

V. L. Makarov and A. M. Rubinov, "Mathematical Theory of Economic Dynamics and Equilibria," Springer-Verlag, New York-Heidelberg, 1977.

[22]

M. Marcus and A. J. Zaslavski, The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 16 (1999), 593-629. doi: 10.1016/S0294-1449(99)80029-8.

[23]

L. W. McKenzie, Turnpike theory, Econometrica, 44 (1976), 841-865. doi: 10.2307/1911532.

[24]

L. W. McKenzie, "Classical General Equilibrium Theory," MIT press, Cambridge, MA, 2002.

[25]

B. Sh. Mordukhovich, Minimax sythesis of a class of control systems with distributed parameters, Automat. Remote Control, 50 (1989), 1333-1340.

[26]

B. S. Mordukhovich and I. Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations, in "Optimal Control, Stabilization and Nonsmooth Analysis," Lecture Notes Control Inform. Sci., 301, Springer, Berlin, (2004), 121-132. doi: 10.1007/978-3-540-39983-4_8.

[27]

S. Pickenhain, V. Lykina and M. Wagner, On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems, Control Cybernet, 37 (2008), 451-468.

[28]

T. Prieto-Rumeau and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games, Math. Methods Oper. Res., 61 (2005), 437-454. doi: 10.1007/s001860400392.

[29]

P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule, American Economic Review, 55 (1965), 486-496.

[30]

C. C. von Weizsacker, Existence of optimal programs of accumulation for an infinite horizon, Rev. Econ. Studies, 32 (1965), 85-104.

[31]

A. J. Zaslavski, Ground states in a model of Frenkel-Kontorova type, Math. USSR-Izvestiya, 29 (1987), 323-354. doi: 10.1070/IM1987v029n02ABEH000972.

[32]

A. J. Zaslavski, Optimal programs on infinite horizon. I, II, SIAM Journal on Control and Optimization, 3 (1995), 1643-1660, 1661-1686. doi: 10.1137/S036301299325726X.

[33]

A. J. Zaslavski, Dynamic properties of optimal solutions of variational problems, Nonlinear Analysis, 27 (1996), 895-931. doi: 10.1016/0362-546X(95)00029-U.

[34]

A. J. Zaslavski, Turnpike property for dynamic discrete time zero-sum games, Abstract and Applied Analysis, 4 (1999), 21-48. doi: 10.1155/S1085337599000020.

[35]

A. J. Zaslavski, "Turnpike Properties in the Calculus of Variations and Optimal Control," Nonconvex Optimization and its Applications, 80, Springer, New York, 2006 .

[36]

A. J. Zaslavski, A turnpike result for a class of problems of the calculus of variations with extended-valued integrands, J. Convex Analysis, 15 (2008), 869-890.

[37]

A. J. Zaslavski, "Optimization on Metric and Normed Spaces," Springer Optimization and Its Applications, 44, Springer, New York, 2010. doi: 10.1007/978-0-387-88621-3.

[38]

A. J. Zaslavski, The existence and structure of approximate solutions of dynamic discrete time zero-sum games, Journal of Nonlinear and Convex Analysis, 12 (2011), 49-68.

[39]

A. J. Zaslavski and A. Leizarowitz, Optimal solutions of linear control systems with nonperiodic integrands, Mathematics of Operations Research, 22 (1997), 726-746. doi: 10.1287/moor.22.3.726.

[1]

Alexander J. Zaslavski. Turnpike properties of approximate solutions of dynamic discrete time zero-sum games. Journal of Dynamics and Games, 2014, 1 (2) : 299-330. doi: 10.3934/jdg.2014.1.299

[2]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001

[3]

Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics and Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020

[4]

Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zero-sum games. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3901-3931. doi: 10.3934/dcds.2015.35.3901

[5]

Zhongbao Zhou, Yanfei Bai, Helu Xiao, Xu Chen. A non-zero-sum reinsurance-investment game with delay and asymmetric information. Journal of Industrial and Management Optimization, 2021, 17 (2) : 909-936. doi: 10.3934/jimo.2020004

[6]

Chloé Jimenez. A zero sum differential game with correlated informations on the initial position. A case with a continuum of initial positions. Journal of Dynamics and Games, 2021, 8 (3) : 233-266. doi: 10.3934/jdg.2021009

[7]

Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control and Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013

[8]

Sylvain Sorin, Guillaume Vigeral. Reversibility and oscillations in zero-sum discounted stochastic games. Journal of Dynamics and Games, 2015, 2 (1) : 103-115. doi: 10.3934/jdg.2015.2.103

[9]

Antoine Hochart. An accretive operator approach to ergodic zero-sum stochastic games. Journal of Dynamics and Games, 2019, 6 (1) : 27-51. doi: 10.3934/jdg.2019003

[10]

Zhi-Wei Sun. Unification of zero-sum problems, subset sums and covers of Z. Electronic Research Announcements, 2003, 9: 51-60.

[11]

Qingmeng Wei, Zhiyong Yu. Time-inconsistent recursive zero-sum stochastic differential games. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1051-1079. doi: 10.3934/mcrf.2018045

[12]

Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics and Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002

[13]

Tao Li, Suresh P. Sethi. A review of dynamic Stackelberg game models. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 125-159. doi: 10.3934/dcdsb.2017007

[14]

Ido Polak, Nicolas Privault. A stochastic newsvendor game with dynamic retail prices. Journal of Industrial and Management Optimization, 2018, 14 (2) : 731-742. doi: 10.3934/jimo.2017072

[15]

Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics and Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013

[16]

Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control and Related Fields, 2022, 12 (2) : 343-370. doi: 10.3934/mcrf.2021025

[17]

Fernando Luque-Vásquez, J. Adolfo Minjárez-Sosa. Average optimal strategies for zero-sum Markov games with poorly known payoff function on one side. Journal of Dynamics and Games, 2014, 1 (1) : 105-119. doi: 10.3934/jdg.2014.1.105

[18]

Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control and Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026

[19]

Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial and Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95

[20]

Fabien Gensbittel, Miquel Oliu-Barton, Xavier Venel. Existence of the uniform value in zero-sum repeated games with a more informed controller. Journal of Dynamics and Games, 2014, 1 (3) : 411-445. doi: 10.3934/jdg.2014.1.411

 Impact Factor: 

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]