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

1. 

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

Received  August 2013 Revised  December 2013 Published  March 2014

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.
Citation: Alexander J. Zaslavski. Turnpike properties of approximate solutions of dynamic discrete time zero-sum games. Journal of Dynamics & 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, 9 (2007), 131.  doi: 10.1007/978-0-8176-4553-3_7.  Google Scholar

[2]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley Interscience, (1984).   Google Scholar

[3]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I,, Physica D, 8 (1983), 381.  doi: 10.1016/0167-2789(83)90233-6.  Google Scholar

[4]

M. Bardi, On differential games with long-time-average cost,, in Advances in Dynamic Games and their Applications, 10 (2009), 3.   Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1137/070696209.  Google Scholar

[8]

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

[9]

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

[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.  doi: 10.1137/S0363012999361962.  Google Scholar

[11]

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

[12]

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

[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.  doi: 10.1016/j.jmaa.2007.08.008.  Google Scholar

[14]

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

[15]

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

[16]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions,, Appl. Analysis, 90 (2011), 1075.  doi: 10.1080/00036811003735840.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s001860400392.  Google Scholar

[20]

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

[21]

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

[22]

A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control,, Springer, (2006).   Google Scholar

[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.   Google Scholar

[24]

A. J. Zaslavski, Structure of Solutions of Variational Problems,, SpringerBriefs in Optimization, (2013).  doi: 10.1007/978-1-4614-6387-0.  Google Scholar

[25]

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

show all references

References:
[1]

O. Alvarez and M. Bardi, Ergodic problems in differential games,, in Advances in Dynamic Game Theory, 9 (2007), 131.  doi: 10.1007/978-0-8176-4553-3_7.  Google Scholar

[2]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley Interscience, (1984).   Google Scholar

[3]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I,, Physica D, 8 (1983), 381.  doi: 10.1016/0167-2789(83)90233-6.  Google Scholar

[4]

M. Bardi, On differential games with long-time-average cost,, in Advances in Dynamic Games and their Applications, 10 (2009), 3.   Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1137/070696209.  Google Scholar

[8]

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

[9]

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

[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.  doi: 10.1137/S0363012999361962.  Google Scholar

[11]

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

[12]

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

[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.  doi: 10.1016/j.jmaa.2007.08.008.  Google Scholar

[14]

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

[15]

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

[16]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions,, Appl. Analysis, 90 (2011), 1075.  doi: 10.1080/00036811003735840.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s001860400392.  Google Scholar

[20]

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

[21]

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

[22]

A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control,, Springer, (2006).   Google Scholar

[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.   Google Scholar

[24]

A. J. Zaslavski, Structure of Solutions of Variational Problems,, SpringerBriefs in Optimization, (2013).  doi: 10.1007/978-1-4614-6387-0.  Google Scholar

[25]

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

[1]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[2]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[3]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[4]

Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300

[5]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[6]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[7]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[8]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[9]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119

[10]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[11]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[12]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275

[13]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[14]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[15]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[16]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[17]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

 Impact Factor: 

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]