Nonzero-sum stochastic differential games with additive structure and average payoffs

Beatris Adriana Escobedo-Trujillo; José Daniel López-Barrientos

doi:10.3934/jdg.2014.1.555

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October 2014, 1(4): 555-578. doi: 10.3934/jdg.2014.1.555

Nonzero-sum stochastic differential games with additive structure and average payoffs

Beatris Adriana Escobedo-Trujillo ^1, and José Daniel López-Barrientos ^2,

1.	Engineering Faculty, Universidad Veracruzana, Coatzacoalcos, Ver. 96538, Mexico
2.	Faculty of Actuarial Sciences, Universidad Anáhuac, Av. Universidad Anáhuac 46. Col. Lomas Anáhuac, Huixquilucan, Edo. de México, Mexico

Received June 2014 Revised September 2014 Published November 2014

Abstract
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This paper deals with nonzero-sum stochastic differential games with an additive structure and long-run average payoffs. Our main objective is to give conditions for the existence of Nash equilibria in the set of relaxed stationary strategies. Such conditions also ensure the existence of a Nash equilibrium within the set of stationary Markov (deterministic) strategies, and that the values of the average payoffs for these equilibria coincide almost everywhere with respect to Lebesgue's measure. This fact generalizes the results in the controlled (single player game) case found by Raghavan [47] and Rosenblueth [48]. We use relaxation theory and standard dynamic programming techniques to achieve our goals. We illustrate our results with an example motivated by a manufacturing system.

Keywords: relaxation theory, Additive structure, average (or ergodic) payoff criterion, dynamic programming, nonzero-sum stochastic differential games..

Mathematics Subject Classification: Primary: 91A10, 91A15, 91A25, 60H1.

Citation: 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

References:

[1]	R. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar
[2]	R. Akella and P. Kumar, Optimal control of production rate in a failure prone manufacturing system, IEEE Trans. Automatic Control, 31 (1986), 116-126. doi: 10.1109/TAC.1986.1104206. Google Scholar
[3]	A. Arapostathis and V. Borkar, Uniform recurrence properties of controlled diffusions and applications to optimal control, SIAM J. Control Optim, 48 (2010), 4181-4223. doi: 10.1137/090762464. Google Scholar
[4]	E. Balder, Lectures on Young Measures, Cahier de Mathématiques de la Décision, CEREMADE, Université Paris IX-Dauphine, Paris 1995. Google Scholar
[5]	M. Bardi, Explicit solutions of some linear-quadratic mean field games, Netw. Heterog. Media., 7 (2012), 243-261. doi: 10.3934/nhm.2012.7.243. Google Scholar
[6]	M. Bardi and F. Priuli, LQG Mean-Field Games with ergodic cost, Proc. 52nd IEEE Conference on Decision and Control, (2013), 2493-2498. doi: 10.1109/CDC.2013.6760255. Google Scholar
[7]	A. Bhatt and R. Karandikar, Invariant measures and evolution equations for Markov processes, Ann. Probab., 21 (1993), 2246-2268. doi: 10.1214/aop/1176989019. Google Scholar
[8]	P. Billingsley, Probability and Measure, $3^{rd}$ edition, Wiley Ser. Probab. Math. Statist., John Willey & Sons, Inc., New York, 1995. Google Scholar
[9]	V. Bogachev, N. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. PartialDifferential Equations, 26 (2001), 2037-2080. doi: 10.1081/PDE-100107815. Google Scholar
[10]	V. Bogachev, N. Krylov and M. Röckner, Regularity and global bounds for the densities of invariant measures of diffusion processes, Dokl. Akad. Nauk, 405 (2005), 583-587. Google Scholar
[11]	V. Bogachev, M. Röckner and V. Stannat, Uniqueness of solutions of elliptic equations and uniqueness pf invariant measures of diffusions processes, Sb. Math., 193 (2002), 945-976. doi: 10.1070/SM2002v193n07ABEH000665. Google Scholar
[12]	V. Borkar, A topology for Markov controls, Appl. Math. Optim., 20 (1989), 55-62. doi: 10.1007/BF01447645. Google Scholar
[13]	V. Borkar and M. Ghosh, Stochastic differential games: Occupation measure based approach, J. Optim. Theory Appl., 73 (1992), 359-385. Correction: 88 (1996), 251-252. doi: 10.1007/BF00940187. Google Scholar
[14]	R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic diferential games, SIAM J. Control Optim., 43 (2004), 624-642. doi: 10.1137/S0363012902411556. Google Scholar
[15]	A. Calderón and J. Rosenblueth, Minimizing Approximate Original Solutions for Commensurate Delayed Controls, Appl. Math. Lett., 7 (1994), 5-10. doi: 10.1016/0893-9659(94)90063-9. Google Scholar
[16]	R. Durrett, Stochastic Calculus: A Practical Introduction, CRC Press, Boca Raton, FLA., 1996. Google Scholar
[17]	A. Dvoretzky, A. Wald and J. Wolfowitz, Elimination of randomization in certain statistical decision procedure and zero-sum two person games, Ann. Math. Statist., 22 (1951), 1-21. doi: 10.1214/aoms/1177729689. Google Scholar
[18]	B. Escobedo-Trujillo, J. López-Barrientos and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum stochastic differential games, J. Optim. Theory Appl., 153 (2012), 662-687. doi: 10.1007/s10957-011-9974-4. Google Scholar
[19]	K. Fan, Fixed-point and minimax theorems in locally convex linear spaces, Proc. N.A.S.U.S.A., 38 (1952), 121-126. doi: 10.1073/pnas.38.2.121. Google Scholar
[20]	K. Fan, Minimax theorems, Proc. N.A.S.U.S.A., 39 (1953), 32-47. doi: 10.1073/pnas.39.1.42. Google Scholar
[21]	G. Folland, Real Analysis. Modern Techniques and Their Applications, $2^{nd}$ edition, John Wiley and Sons, New York, 1999. Google Scholar
[22]	A. Friedman, Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York, 1975. Google Scholar
[23]	M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions, SIAM J. Control Optim., 35 (1997), 1962-1988. doi: 10.1137/S0363012996299302. Google Scholar
[24]	M. Ghosh and A. Bagchi, Stochastic game with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301. doi: 10.1007/s002459900092. Google Scholar
[25]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprinted version, Springer, Heidelberg, 1998. Google Scholar
[26]	P. Glynn and S. Meyn, A Liapounov bound for solutions of the Poisson equation, Ann. Prob., 24 (1996), 916-931. doi: 10.1214/aop/1039639370. Google Scholar
[27]	J. González-Hernández and O. Hernández-Lerma, Extreme points of sets of randomized strategies in constrained optimization and control problems, SIAM J. Optim., 15 (2005), 1085-1104. doi: 10.1137/040605345. Google Scholar
[28]	H. Hernández-Hernández, Existence of nash equilibria in discounted nonzero-sum stochastic games with additive structure, Morfismos, 6 (2002), 43-65. Google Scholar
[29]	O. Hernández-Lerma and J. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer, New York, 1996. doi: 10.1007/978-1-4612-0729-0. Google Scholar
[30]	O. Hernández-Lerma and J. Lasserre, Further topics on Discrete-Time Markov Control Processes, Springer, New York, 1999. doi: 10.1007/978-1-4612-0561-6. Google Scholar
[31]	H. Jasso-Fuentes and O. Hernández-Lerma, Characterizations of overtaking optimality for controlled diffusion processes, Appl. Math. Optim., 57 (2008), 349-369. doi: 10.1007/s00245-007-9025-6. Google Scholar
[32]	F. Klebaner, Introduction to Stochastic Calculus with Applications, $2^{nd}$ edition, Imperial College Press, London, 2005. doi: 10.1142/p386. Google Scholar
[33]	A. Knapp, Advanced Real Analysis, $2^{nd}$ edition, Birkäuser, New York, 2005. Google Scholar
[34]	A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover Publications, 1999. Google Scholar
[35]	N. Krylov, Controlled diffusion processes. Applications of Mathematics, Springer-Verlag, New York, 1980. Google Scholar
[36]	H. Kushner, Numerical approximations for nonzero-sum stochastic differential games, SIAM J.Control Optim., 46 (2007), 1942-1971. doi: 10.1137/050647931. Google Scholar
[37]	H. Küenle, Equilibrium strategies in stochastic games with additive cost and transition structure, Internat. Game Theory Rev., 1 (1999), 131-147. doi: 10.1142/S0219198999000098. Google Scholar
[38]	S. Meyn and R. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time precesses, Adv. Appl. Prob., 25 (1993), 518-548. doi: 10.2307/1427522. Google Scholar
[39]	H. Morimoto and M. Ohashi, On linear stochastic differential games with average cost criterions, J. Optimization Theory and Applications, 64 (1990), 127-140. doi: 10.1007/BF00940027. Google Scholar
[40]	J. Munkres, Topology, $2^{nd}$ edition, Prentice Hall Inc., 2000. Google Scholar
[41]	A. Nowak, Stationary equilibria for nonzero-sum average payoff ergodic stochastic games with general state space, Advances in Dynamic Games and Applications (T. Basar and A. Haurie, eds.), Birkhäuser, 1 (1994), 231-246. Google Scholar
[42]	A. Nowak, Remark on sensitive equilibria in stochastic games with additive reward and transition structure, Math. Meth. Oper. Res., 64 (2006), 481-494. doi: 10.1007/s00186-006-0090-4. Google Scholar
[43]	P. Pedregal, Optimization, relaxation and Young measures, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 27-58. doi: 10.1090/S0273-0979-99-00774-0. Google Scholar
[44]	B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, $4^{th}$ edition, Springer, New York, 1995. doi: 10.1007/978-3-662-03185-8. Google Scholar
[45]	K. Parthasarathy and T. Steerneman, A tool in establishing total variation convergence, Proceedings of the American Mathematical Society, 95 (1985), 626-630. doi: 10.1090/S0002-9939-1985-0810175-X. Google Scholar
[46]	T. Prieto-Rumeau and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games, Math. Meth. Oper. Res., 61 (2005), 437-454. doi: 10.1007/s001860400392. Google Scholar
[47]	T. Raghavan, S. Tijs and O. Vrieze, On stochastic games with additive reward and transition structure, J. Optim. Theory Appl., 47 (1985), 451-464. doi: 10.1007/BF00942191. Google Scholar
[48]	J. Rosenblueth, Proper relaxation of optimal control problem, J. Optim. Theory Appl., 74 (1992), 509-526. doi: 10.1007/BF00940324. Google Scholar
[49]	N. Saldi, T. Linder and S. Yüksel, Asymptotic optimality of quantized control in Markov decision processes., Submitted to IEEE Conference on Decision and Control., (). Google Scholar
[50]	M. Schäl, Conditions for optimality and for the limit of $n-$stage optimal policies to be optimal, Z. Wahrs. Verw. Gerb., 32 (1975), 179-196. doi: 10.1007/BF00532612. Google Scholar
[51]	L. Young, Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders, Philadelphia, PA, 1969. Google Scholar
[52]	J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. Google Scholar
[53]	E. Zeidler, Nonlinear Functional Analysis and its Applications II/A, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0. Google Scholar

show all references

References:

[1]	R. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar
[2]	R. Akella and P. Kumar, Optimal control of production rate in a failure prone manufacturing system, IEEE Trans. Automatic Control, 31 (1986), 116-126. doi: 10.1109/TAC.1986.1104206. Google Scholar
[3]	A. Arapostathis and V. Borkar, Uniform recurrence properties of controlled diffusions and applications to optimal control, SIAM J. Control Optim, 48 (2010), 4181-4223. doi: 10.1137/090762464. Google Scholar
[4]	E. Balder, Lectures on Young Measures, Cahier de Mathématiques de la Décision, CEREMADE, Université Paris IX-Dauphine, Paris 1995. Google Scholar
[5]	M. Bardi, Explicit solutions of some linear-quadratic mean field games, Netw. Heterog. Media., 7 (2012), 243-261. doi: 10.3934/nhm.2012.7.243. Google Scholar
[6]	M. Bardi and F. Priuli, LQG Mean-Field Games with ergodic cost, Proc. 52nd IEEE Conference on Decision and Control, (2013), 2493-2498. doi: 10.1109/CDC.2013.6760255. Google Scholar
[7]	A. Bhatt and R. Karandikar, Invariant measures and evolution equations for Markov processes, Ann. Probab., 21 (1993), 2246-2268. doi: 10.1214/aop/1176989019. Google Scholar
[8]	P. Billingsley, Probability and Measure, $3^{rd}$ edition, Wiley Ser. Probab. Math. Statist., John Willey & Sons, Inc., New York, 1995. Google Scholar
[9]	V. Bogachev, N. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. PartialDifferential Equations, 26 (2001), 2037-2080. doi: 10.1081/PDE-100107815. Google Scholar
[10]	V. Bogachev, N. Krylov and M. Röckner, Regularity and global bounds for the densities of invariant measures of diffusion processes, Dokl. Akad. Nauk, 405 (2005), 583-587. Google Scholar
[11]	V. Bogachev, M. Röckner and V. Stannat, Uniqueness of solutions of elliptic equations and uniqueness pf invariant measures of diffusions processes, Sb. Math., 193 (2002), 945-976. doi: 10.1070/SM2002v193n07ABEH000665. Google Scholar
[12]	V. Borkar, A topology for Markov controls, Appl. Math. Optim., 20 (1989), 55-62. doi: 10.1007/BF01447645. Google Scholar
[13]	V. Borkar and M. Ghosh, Stochastic differential games: Occupation measure based approach, J. Optim. Theory Appl., 73 (1992), 359-385. Correction: 88 (1996), 251-252. doi: 10.1007/BF00940187. Google Scholar
[14]	R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic diferential games, SIAM J. Control Optim., 43 (2004), 624-642. doi: 10.1137/S0363012902411556. Google Scholar
[15]	A. Calderón and J. Rosenblueth, Minimizing Approximate Original Solutions for Commensurate Delayed Controls, Appl. Math. Lett., 7 (1994), 5-10. doi: 10.1016/0893-9659(94)90063-9. Google Scholar
[16]	R. Durrett, Stochastic Calculus: A Practical Introduction, CRC Press, Boca Raton, FLA., 1996. Google Scholar
[17]	A. Dvoretzky, A. Wald and J. Wolfowitz, Elimination of randomization in certain statistical decision procedure and zero-sum two person games, Ann. Math. Statist., 22 (1951), 1-21. doi: 10.1214/aoms/1177729689. Google Scholar
[18]	B. Escobedo-Trujillo, J. López-Barrientos and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum stochastic differential games, J. Optim. Theory Appl., 153 (2012), 662-687. doi: 10.1007/s10957-011-9974-4. Google Scholar
[19]	K. Fan, Fixed-point and minimax theorems in locally convex linear spaces, Proc. N.A.S.U.S.A., 38 (1952), 121-126. doi: 10.1073/pnas.38.2.121. Google Scholar
[20]	K. Fan, Minimax theorems, Proc. N.A.S.U.S.A., 39 (1953), 32-47. doi: 10.1073/pnas.39.1.42. Google Scholar
[21]	G. Folland, Real Analysis. Modern Techniques and Their Applications, $2^{nd}$ edition, John Wiley and Sons, New York, 1999. Google Scholar
[22]	A. Friedman, Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York, 1975. Google Scholar
[23]	M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions, SIAM J. Control Optim., 35 (1997), 1962-1988. doi: 10.1137/S0363012996299302. Google Scholar
[24]	M. Ghosh and A. Bagchi, Stochastic game with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301. doi: 10.1007/s002459900092. Google Scholar
[25]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprinted version, Springer, Heidelberg, 1998. Google Scholar
[26]	P. Glynn and S. Meyn, A Liapounov bound for solutions of the Poisson equation, Ann. Prob., 24 (1996), 916-931. doi: 10.1214/aop/1039639370. Google Scholar
[27]	J. González-Hernández and O. Hernández-Lerma, Extreme points of sets of randomized strategies in constrained optimization and control problems, SIAM J. Optim., 15 (2005), 1085-1104. doi: 10.1137/040605345. Google Scholar
[28]	H. Hernández-Hernández, Existence of nash equilibria in discounted nonzero-sum stochastic games with additive structure, Morfismos, 6 (2002), 43-65. Google Scholar
[29]	O. Hernández-Lerma and J. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer, New York, 1996. doi: 10.1007/978-1-4612-0729-0. Google Scholar
[30]	O. Hernández-Lerma and J. Lasserre, Further topics on Discrete-Time Markov Control Processes, Springer, New York, 1999. doi: 10.1007/978-1-4612-0561-6. Google Scholar
[31]	H. Jasso-Fuentes and O. Hernández-Lerma, Characterizations of overtaking optimality for controlled diffusion processes, Appl. Math. Optim., 57 (2008), 349-369. doi: 10.1007/s00245-007-9025-6. Google Scholar
[32]	F. Klebaner, Introduction to Stochastic Calculus with Applications, $2^{nd}$ edition, Imperial College Press, London, 2005. doi: 10.1142/p386. Google Scholar
[33]	A. Knapp, Advanced Real Analysis, $2^{nd}$ edition, Birkäuser, New York, 2005. Google Scholar
[34]	A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover Publications, 1999. Google Scholar
[35]	N. Krylov, Controlled diffusion processes. Applications of Mathematics, Springer-Verlag, New York, 1980. Google Scholar
[36]	H. Kushner, Numerical approximations for nonzero-sum stochastic differential games, SIAM J.Control Optim., 46 (2007), 1942-1971. doi: 10.1137/050647931. Google Scholar
[37]	H. Küenle, Equilibrium strategies in stochastic games with additive cost and transition structure, Internat. Game Theory Rev., 1 (1999), 131-147. doi: 10.1142/S0219198999000098. Google Scholar
[38]	S. Meyn and R. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time precesses, Adv. Appl. Prob., 25 (1993), 518-548. doi: 10.2307/1427522. Google Scholar
[39]	H. Morimoto and M. Ohashi, On linear stochastic differential games with average cost criterions, J. Optimization Theory and Applications, 64 (1990), 127-140. doi: 10.1007/BF00940027. Google Scholar
[40]	J. Munkres, Topology, $2^{nd}$ edition, Prentice Hall Inc., 2000. Google Scholar
[41]	A. Nowak, Stationary equilibria for nonzero-sum average payoff ergodic stochastic games with general state space, Advances in Dynamic Games and Applications (T. Basar and A. Haurie, eds.), Birkhäuser, 1 (1994), 231-246. Google Scholar
[42]	A. Nowak, Remark on sensitive equilibria in stochastic games with additive reward and transition structure, Math. Meth. Oper. Res., 64 (2006), 481-494. doi: 10.1007/s00186-006-0090-4. Google Scholar
[43]	P. Pedregal, Optimization, relaxation and Young measures, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 27-58. doi: 10.1090/S0273-0979-99-00774-0. Google Scholar
[44]	B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, $4^{th}$ edition, Springer, New York, 1995. doi: 10.1007/978-3-662-03185-8. Google Scholar
[45]	K. Parthasarathy and T. Steerneman, A tool in establishing total variation convergence, Proceedings of the American Mathematical Society, 95 (1985), 626-630. doi: 10.1090/S0002-9939-1985-0810175-X. Google Scholar
[46]	T. Prieto-Rumeau and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games, Math. Meth. Oper. Res., 61 (2005), 437-454. doi: 10.1007/s001860400392. Google Scholar
[47]	T. Raghavan, S. Tijs and O. Vrieze, On stochastic games with additive reward and transition structure, J. Optim. Theory Appl., 47 (1985), 451-464. doi: 10.1007/BF00942191. Google Scholar
[48]	J. Rosenblueth, Proper relaxation of optimal control problem, J. Optim. Theory Appl., 74 (1992), 509-526. doi: 10.1007/BF00940324. Google Scholar
[49]	N. Saldi, T. Linder and S. Yüksel, Asymptotic optimality of quantized control in Markov decision processes., Submitted to IEEE Conference on Decision and Control., (). Google Scholar
[50]	M. Schäl, Conditions for optimality and for the limit of $n-$stage optimal policies to be optimal, Z. Wahrs. Verw. Gerb., 32 (1975), 179-196. doi: 10.1007/BF00532612. Google Scholar
[51]	L. Young, Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders, Philadelphia, PA, 1969. Google Scholar
[52]	J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. Google Scholar
[53]	E. Zeidler, Nonlinear Functional Analysis and its Applications II/A, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0. Google Scholar

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