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

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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

Abstract
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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 & 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

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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

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