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 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.

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