Discount-sensitive equilibria in zero-sum stochastic differential games

Previous Article
Uncertainty and inside information
JDG Home
This Issue
Next Article
Similarity solutions of a multidimensional replicator dynamics integrodifferential equation

January 2016, 3(1): 25-50. doi: 10.3934/jdg.2016002

Discount-sensitive equilibria in zero-sum stochastic differential games

1.	Engineering Faculty, Universidad Veracruzana, Coatzacoalcos, Ver. 96538, Mexico

Received April 2015 Revised November 2015 Published March 2016

Abstract
Related Papers

We consider infinite-horizon zero-sum stochastic differential games with average payoff criteria, discount -sensitive criteria and, infinite-horizon undiscounted reward criteria which are sensitive to the growth rate of finite-horizon payoffs. These criteria include, average reward optimality, strong 0-discount optimality, strong -1-discount optimality, 0-discount optimality, bias optimality, F-strong average optimality and overtaking optimality. The main objective is to give conditions under which these criteria are interrelated.

Keywords: strong average., Additive structure, sensitive equilibria, Nash equlibrium, $m-$ discount optimality.

Mathematics Subject Classification: Primary: 91A10, 91A15, 91A2.

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

References:

[1]	A. Arapostathis and V. Borkar, Uniform recurrence properties of controlled diffusions and applications to optimal control,, SIAM J. Control Optim., 48 (2010), 4181. doi: 10.1137/090762464.
[2]	A. Arapostathis, M. Ghosh and V. Borkar, Ergodic Control of Diffusion Processes,, Vol. 143, (2012).
[3]	A. Arapostathis, V. Borkar and K. Surech, Relative value iteration for stochastic differential games,, , 13 (2013), 3. doi: 10.1007/978-3-319-02690-9_1.
[4]	M. Bardi, Explicit solutions of some linear-quadratic mean field games. Networks and heterogeneous media,, American Institute of Mathematical Sciences, 7 (2012), 243. doi: 10.3934/nhm.2012.7.243.
[5]	V. Borkar and M. Ghosh, Stochastic differential games: Occupation measure based approach,, J. Optim. Theory Appl., 73 (1992), 359. doi: 10.1007/BF00940187.
[6]	R. Cavazos-Cadena and J. B. Lasserre, Strong 1-optimal stationary policies in denumerable Markov decision processes,, Syst. Control Lett., 11 (1988), 65. doi: 10.1016/0167-6911(88)90113-2.
[7]	B. Escobedo-Trujillo, D. López-Barrientos and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum stochastic differential games,, J. Optim. Theory Appl., 153 (2012), 662. doi: 10.1007/s10957-011-9974-4.
[8]	B. Escobedo-Trujillo and J. López-Barrientos, Nonzero-sum stochastic differential games with additive structure and average payoffs,, Journal of Dynamics and Games, 1 (2014), 555. doi: 10.3934/jdg.2014.1.555.
[9]	K. Fan, Fixed-point and minimax theorems in locally convex linear spaces,, Proc. N.A.S.U.S.A., 38 (1952), 121. doi: 10.1073/pnas.38.2.121.
[10]	A. Friedman, Stochastic Differential Equations and Applications,, Vol. 1, (1975).
[11]	J. Flynn, On optimality criteria for dynamic programs with long finite horizons,, J. Math. Anal. Appl., 76 (1980), 202. doi: 10.1016/0022-247X(80)90072-4.
[12]	M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1952. doi: 10.1137/S0363012996299302.
[13]	O. Hernández-Lerma and O. Vega-Amaya, Infinite-horizon Markov control processes with undiscounted cost criteria: from average to overtaking optimality,, Appl. Math. (Warsaw), 25 (1998), 153.
[14]	O. Hernández-Lerma and J. Lasserre, Further Topics on Discrete-Time Markov Control Processes,, Springer, (1999). doi: 10.1007/978-1-4612-0561-6.
[15]	N. Hilgert and O. Hernández-Lerma, Bias optimality versus strong 0-discount optimality in Markov control processes with unbounded costs,, Acta Applicandae Mathematicae, 77 (2003), 215. doi: 10.1023/A:1024996308133.
[16]	H. Jasso-Fuentes and O. Hernández-Lerma, Characterizations of overtaking optimality for controlled diffusion processes,, Appl. Math. Optim., 57 (2008), 349. doi: 10.1007/s00245-007-9025-6.
[17]	H. Jasso-Fuentes and O. Hernández-Lerma, Ergodic control, bias, and sensitive discount optimality for Markov diffusion processes,, Stochatic Analysis and Applications, 27 (2009), 363. doi: 10.1080/07362990802679034.
[18]	H. Jasso-Fuentes, J. López-Barrientos and B. Escobedo-Trujilo, Infinite horizon nonzero-sum stochastic differential games with additive structure,, IMA J. Math. Control Inform. doi: 10.1093/imamci/dnv045, (2015).
[19]	S. Meyn and R. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time precesses,, Adv. Appl. Prob., 25 (1993), 518. doi: 10.2307/1427522.
[20]	H. Morimoto and M. Ohashi, On linear stochastic differential games with average cost criterions,, J. Optim. Theory Appl., 64 (1990), 127. doi: 10.1007/BF00940027.
[21]	A. Nowak, Sensitive equilibria for ergodic stochastic games with countable state spaces,, Math. Meth. Oper. Res., 50 (1999), 65.
[22]	A. Nowak, Optimal strategies in a class of zero-sum ergodic stochastic games,, Math. Meth. Oper. Res., 50 (1999), 399. doi: 10.1007/s001860050078.
[23]	A. Nowak, Remark on sensitive equilibria in stochastic games with additive reward and transition structure,, Math. Meth. Oper. Res., 64 (2006), 481. doi: 10.1007/s00186-006-0090-4.
[24]	T. Prieto-Rumeau and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games,, Math. Meth. Oper. Res., 61 (2005), 437. doi: 10.1007/s001860400392.
[25]	T. Prieto-Rumeau and O. Hernández-Lerma, The Laurent series, sensitive discount and Blackwell optimality for continuous-time controlled Markov chains,, Math. Meth. Oper. Res., 61 (2005), 123. doi: 10.1007/s001860400393.
[26]	M. Puterman, Sensitive discount optimality in controlled one-dimensional diffusions,, Annals of Probability, 2 (1974), 408. doi: 10.1214/aop/1176996656.
[27]	W. Qingda and C. Xian, Strong average optimality criterion for continuos-time Markov decision processes,, Kybernetika, 50 (2014), 950.
[28]	M. Sion, On general minimax theorems,, Pacific J. Math., 8 (1958), 171. doi: 10.2140/pjm.1958.8.171.
[29]	W. Schmitendorf, Differential games without pure strategy saddle-point solutions,, J. Optim. Theory Appl., 18 (1976), 81. doi: 10.1007/BF00933796.
[30]	Q. Zhu and X. Guo, Another set of conditions for strong $n$ ($n=-1,0$) discount optimality in Markov decision processes,, Stochastic Analysis and Applications, 23 (2005), 953. doi: 10.1080/07362990500184865.
[31]	Q. Zhu, Bias optimality and strong $n$ ($n=-1,0$) discount optimality for Markov decision processes,, J. Math. Anal. Appl., 334 (2007), 576. doi: 10.1016/j.jmaa.2007.01.002.

show all references

References:

[1]	A. Arapostathis and V. Borkar, Uniform recurrence properties of controlled diffusions and applications to optimal control,, SIAM J. Control Optim., 48 (2010), 4181. doi: 10.1137/090762464.
[2]	A. Arapostathis, M. Ghosh and V. Borkar, Ergodic Control of Diffusion Processes,, Vol. 143, (2012).
[3]	A. Arapostathis, V. Borkar and K. Surech, Relative value iteration for stochastic differential games,, , 13 (2013), 3. doi: 10.1007/978-3-319-02690-9_1.
[4]	M. Bardi, Explicit solutions of some linear-quadratic mean field games. Networks and heterogeneous media,, American Institute of Mathematical Sciences, 7 (2012), 243. doi: 10.3934/nhm.2012.7.243.
[5]	V. Borkar and M. Ghosh, Stochastic differential games: Occupation measure based approach,, J. Optim. Theory Appl., 73 (1992), 359. doi: 10.1007/BF00940187.
[6]	R. Cavazos-Cadena and J. B. Lasserre, Strong 1-optimal stationary policies in denumerable Markov decision processes,, Syst. Control Lett., 11 (1988), 65. doi: 10.1016/0167-6911(88)90113-2.
[7]	B. Escobedo-Trujillo, D. López-Barrientos and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum stochastic differential games,, J. Optim. Theory Appl., 153 (2012), 662. doi: 10.1007/s10957-011-9974-4.
[8]	B. Escobedo-Trujillo and J. López-Barrientos, Nonzero-sum stochastic differential games with additive structure and average payoffs,, Journal of Dynamics and Games, 1 (2014), 555. doi: 10.3934/jdg.2014.1.555.
[9]	K. Fan, Fixed-point and minimax theorems in locally convex linear spaces,, Proc. N.A.S.U.S.A., 38 (1952), 121. doi: 10.1073/pnas.38.2.121.
[10]	A. Friedman, Stochastic Differential Equations and Applications,, Vol. 1, (1975).
[11]	J. Flynn, On optimality criteria for dynamic programs with long finite horizons,, J. Math. Anal. Appl., 76 (1980), 202. doi: 10.1016/0022-247X(80)90072-4.
[12]	M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1952. doi: 10.1137/S0363012996299302.
[13]	O. Hernández-Lerma and O. Vega-Amaya, Infinite-horizon Markov control processes with undiscounted cost criteria: from average to overtaking optimality,, Appl. Math. (Warsaw), 25 (1998), 153.
[14]	O. Hernández-Lerma and J. Lasserre, Further Topics on Discrete-Time Markov Control Processes,, Springer, (1999). doi: 10.1007/978-1-4612-0561-6.
[15]	N. Hilgert and O. Hernández-Lerma, Bias optimality versus strong 0-discount optimality in Markov control processes with unbounded costs,, Acta Applicandae Mathematicae, 77 (2003), 215. doi: 10.1023/A:1024996308133.
[16]	H. Jasso-Fuentes and O. Hernández-Lerma, Characterizations of overtaking optimality for controlled diffusion processes,, Appl. Math. Optim., 57 (2008), 349. doi: 10.1007/s00245-007-9025-6.
[17]	H. Jasso-Fuentes and O. Hernández-Lerma, Ergodic control, bias, and sensitive discount optimality for Markov diffusion processes,, Stochatic Analysis and Applications, 27 (2009), 363. doi: 10.1080/07362990802679034.
[18]	H. Jasso-Fuentes, J. López-Barrientos and B. Escobedo-Trujilo, Infinite horizon nonzero-sum stochastic differential games with additive structure,, IMA J. Math. Control Inform. doi: 10.1093/imamci/dnv045, (2015).
[19]	S. Meyn and R. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time precesses,, Adv. Appl. Prob., 25 (1993), 518. doi: 10.2307/1427522.
[20]	H. Morimoto and M. Ohashi, On linear stochastic differential games with average cost criterions,, J. Optim. Theory Appl., 64 (1990), 127. doi: 10.1007/BF00940027.
[21]	A. Nowak, Sensitive equilibria for ergodic stochastic games with countable state spaces,, Math. Meth. Oper. Res., 50 (1999), 65.
[22]	A. Nowak, Optimal strategies in a class of zero-sum ergodic stochastic games,, Math. Meth. Oper. Res., 50 (1999), 399. doi: 10.1007/s001860050078.
[23]	A. Nowak, Remark on sensitive equilibria in stochastic games with additive reward and transition structure,, Math. Meth. Oper. Res., 64 (2006), 481. doi: 10.1007/s00186-006-0090-4.
[24]	T. Prieto-Rumeau and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games,, Math. Meth. Oper. Res., 61 (2005), 437. doi: 10.1007/s001860400392.
[25]	T. Prieto-Rumeau and O. Hernández-Lerma, The Laurent series, sensitive discount and Blackwell optimality for continuous-time controlled Markov chains,, Math. Meth. Oper. Res., 61 (2005), 123. doi: 10.1007/s001860400393.
[26]	M. Puterman, Sensitive discount optimality in controlled one-dimensional diffusions,, Annals of Probability, 2 (1974), 408. doi: 10.1214/aop/1176996656.
[27]	W. Qingda and C. Xian, Strong average optimality criterion for continuos-time Markov decision processes,, Kybernetika, 50 (2014), 950.
[28]	M. Sion, On general minimax theorems,, Pacific J. Math., 8 (1958), 171. doi: 10.2140/pjm.1958.8.171.
[29]	W. Schmitendorf, Differential games without pure strategy saddle-point solutions,, J. Optim. Theory Appl., 18 (1976), 81. doi: 10.1007/BF00933796.
[30]	Q. Zhu and X. Guo, Another set of conditions for strong $n$ ($n=-1,0$) discount optimality in Markov decision processes,, Stochastic Analysis and Applications, 23 (2005), 953. doi: 10.1080/07362990500184865.
[31]	Q. Zhu, Bias optimality and strong $n$ ($n=-1,0$) discount optimality for Markov decision processes,, J. Math. Anal. Appl., 334 (2007), 576. doi: 10.1016/j.jmaa.2007.01.002.

[1]	Beatris Adriana Escobedo-Trujillo, Alejandro Alaffita-Hernández, Raquiel López-Martínez. Constrained stochastic differential games with additive structure: Average and discount payoffs. Journal of Dynamics & Games, 2018, 5 (2) : 109-141. doi: 10.3934/jdg.2018008
[2]	Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos. Nonzero-sum stochastic differential games with additive structure and average payoffs. Journal of Dynamics & Games, 2014, 1 (4) : 555-578. doi: 10.3934/jdg.2014.1.555
[3]	Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537
[4]	Sheri M. Markose. Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations. Journal of Dynamics & Games, 2017, 4 (3) : 255-284. doi: 10.3934/jdg.2017015
[5]	Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086
[6]	Margarida Carvalho, João Pedro Pedroso, João Saraiva. Electricity day-ahead markets: Computation of Nash equilibria. Journal of Industrial & Management Optimization, 2015, 11 (3) : 985-998. doi: 10.3934/jimo.2015.11.985
[7]	Matthew Bourque, T. E. S. Raghavan. Policy improvement for perfect information additive reward and additive transition stochastic games with discounted and average payoffs. Journal of Dynamics & Games, 2014, 1 (3) : 347-361. doi: 10.3934/jdg.2014.1.347
[8]	Mei Ju Luo, Yi Zeng Chen. Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 1-15. doi: 10.3934/jimo.2016.12.1
[9]	Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models^†. Journal of Dynamics & Games, 2017, 4 (3) : 195-203. doi: 10.3934/jdg.2017012
[10]	Adela Capătă. Optimality conditions for strong vector equilibrium problems under a weak constraint qualification. Journal of Industrial & Management Optimization, 2015, 11 (2) : 563-574. doi: 10.3934/jimo.2015.11.563
[11]	Zilong Wang, Guang Gong, Rongquan Feng. A generalized construction of OFDM $M$-QAM sequences with low peak-to-average power ratio. Advances in Mathematics of Communications, 2009, 3 (4) : 421-428. doi: 10.3934/amc.2009.3.421
[12]	Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[13]	Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136
[14]	Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657
[15]	Jerim Kim, Bara Kim, Hwa-Sung Kim. G/M/1 type structure of a risk model with general claim sizes in a Markovian environment. Journal of Industrial & Management Optimization, 2012, 8 (4) : 909-924. doi: 10.3934/jimo.2012.8.909
[16]	Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial & Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609
[17]	Janos Kollar. The Nash conjecture for threefolds. Electronic Research Announcements, 1998, 4: 63-73.
[18]	William Geller, Bruce Kitchens, Michał Misiurewicz. Microdynamics for Nash maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1007-1024. doi: 10.3934/dcds.2010.27.1007
[19]	Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[20]	Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial & Management Optimization, 2009, 5 (3) : 431-451. doi: 10.3934/jimo.2009.5.431

Impact Factor:

American Institute of Mathematical Sciences