Turnpike properties of approximate solutions of dynamic discrete time zero-sum games

Previous Article
JDG Home
This Issue
Next Article
A strategic market game approach for the private provision of public goods

April 2014, 1(2): 299-330. doi: 10.3934/jdg.2014.1.299

Turnpike properties of approximate solutions of dynamic discrete time zero-sum games

Alexander J. Zaslavski ^1,

1.	Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Received August 2013 Revised December 2013 Published March 2014

Abstract
Related Papers

We study existence and turnpike properties of approximate solutions for a class of dynamic discrete-time two-player zero-sum games without using convexity-concavity assumptions. We describe the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals and show that approximate solutions are determined mainly by the objective function, and are essentially independent of the choice of interval and endpoint conditions.

Keywords: Approximate solution, zero-sum game., dynamic game, turnpike property.

Mathematics Subject Classification: Primary: 49J99, 91A05, 91A2.

Citation: 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

References:

[1]	O. Alvarez and M. Bardi, Ergodic problems in differential games, in Advances in Dynamic Game Theory, Birkhäuser, 9 (2007), 131-152. doi: 10.1007/978-0-8176-4553-3_7.
[2]	J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley Interscience, New York, 1984.
[3]	S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I, Physica D, 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.
[4]	M. Bardi, On differential games with long-time-average cost, in Advances in Dynamic Games and their Applications, Birkhäuser, 10 (2009), 3-18.
[5]	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.
[6]	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.
[7]	V. Gaitsgory and M. Quincampoix, Linear programming approach to deterministic infinite horizon optimal control problems with discounting, SIAM J. Control and Optimization, 48 (2009), 2480-2512. doi: 10.1137/070696209.
[8]	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.
[9]	H. Jasso-Fuentes and O. Hernandez-Lerma, Characterizations of overtaking optimality for controlled diffusion processes, Appl. Math. Optim., 57 (2008), 349-369. doi: 10.1007/s00245-007-9025-6.
[10]	O. Hernandez-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.
[11]	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.
[12]	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.
[13]	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.
[14]	M. Marcus and A. J. Zaslavski, The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincare, Anal. Non Lineaire, 16 (1999), 593-629. doi: 10.1016/S0294-1449(99)80029-8.
[15]	L. W. McKenzie, Turnpike theory, Econometrica, 44 (1976), 841-865. doi: 10.2307/1911532.
[16]	B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions, Appl. Analysis, 90 (2011), 1075-1109. doi: 10.1080/00036811003735840.
[17]	E. Ocana Anaya, P. Cartigny and P. Loisel, Singular infinite horizon calculus of variations. Applications to fisheries management, J. Nonlinear Convex Anal., 10 (2009), 157-176.
[18]	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.
[19]	T. Prieto-Rumeau and O. Hernandez-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games, Math. Methods Oper. Res., 61 (2005), 437-454. doi: 10.1007/s001860400392.
[20]	P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule, American Economic Review, 55 (1965), 486-496.
[21]	A. J. Zaslavski, Turnpike property for dynamic discrete time zero-sum games, Abstract and Applied Analysis, 4 (1999), 21-48. doi: 10.1155/S1085337599000020.
[22]	A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006.
[23]	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.
[24]	A. J. Zaslavski, Structure of Solutions of Variational Problems, SpringerBriefs in Optimization, New York, 2013. doi: 10.1007/978-1-4614-6387-0.
[25]	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, Birkhäuser, 9 (2007), 131-152. doi: 10.1007/978-0-8176-4553-3_7.
[2]	J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley Interscience, New York, 1984.
[3]	S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I, Physica D, 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.
[4]	M. Bardi, On differential games with long-time-average cost, in Advances in Dynamic Games and their Applications, Birkhäuser, 10 (2009), 3-18.
[5]	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.
[6]	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.
[7]	V. Gaitsgory and M. Quincampoix, Linear programming approach to deterministic infinite horizon optimal control problems with discounting, SIAM J. Control and Optimization, 48 (2009), 2480-2512. doi: 10.1137/070696209.
[8]	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.
[9]	H. Jasso-Fuentes and O. Hernandez-Lerma, Characterizations of overtaking optimality for controlled diffusion processes, Appl. Math. Optim., 57 (2008), 349-369. doi: 10.1007/s00245-007-9025-6.
[10]	O. Hernandez-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.
[11]	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.
[12]	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.
[13]	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.
[14]	M. Marcus and A. J. Zaslavski, The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincare, Anal. Non Lineaire, 16 (1999), 593-629. doi: 10.1016/S0294-1449(99)80029-8.
[15]	L. W. McKenzie, Turnpike theory, Econometrica, 44 (1976), 841-865. doi: 10.2307/1911532.
[16]	B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions, Appl. Analysis, 90 (2011), 1075-1109. doi: 10.1080/00036811003735840.
[17]	E. Ocana Anaya, P. Cartigny and P. Loisel, Singular infinite horizon calculus of variations. Applications to fisheries management, J. Nonlinear Convex Anal., 10 (2009), 157-176.
[18]	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.
[19]	T. Prieto-Rumeau and O. Hernandez-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games, Math. Methods Oper. Res., 61 (2005), 437-454. doi: 10.1007/s001860400392.
[20]	P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule, American Economic Review, 55 (1965), 486-496.
[21]	A. J. Zaslavski, Turnpike property for dynamic discrete time zero-sum games, Abstract and Applied Analysis, 4 (1999), 21-48. doi: 10.1155/S1085337599000020.
[22]	A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006.
[23]	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.
[24]	A. J. Zaslavski, Structure of Solutions of Variational Problems, SpringerBriefs in Optimization, New York, 2013. doi: 10.1007/978-1-4614-6387-0.
[25]	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. 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
[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]	Abd El-Monem A. Megahed, Ebrahim A. Youness, Hebatallah K. Arafat. Optimization method in counter terrorism: Min-Max zero-sum differential game approach. Numerical Algebra, Control and Optimization, 2022 doi: 10.3934/naco.2022013
[4]	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
[5]	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
[6]	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
[7]	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
[8]	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
[9]	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
[10]	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
[11]	Zhi-Wei Sun. Unification of zero-sum problems, subset sums and covers of Z. Electronic Research Announcements, 2003, 9: 51-60.
[12]	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
[13]	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
[14]	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
[15]	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
[16]	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
[17]	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
[18]	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
[19]	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
[20]	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

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Alexander J. Zaslavski
on Google Scholar
Alexander J. Zaslavski

[Back to Top]