# American Institute of Mathematical Sciences

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

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

 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

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.
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, (1975). [2] R. Akella and P. Kumar, Optimal control of production rate in a failure prone manufacturing system,, IEEE Trans. Automatic Control, 31 (1986), 116. doi: 10.1109/TAC.1986.1104206. [3] 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. [4] E. Balder, Lectures on Young Measures,, Cahier de Mathématiques de la Décision, (1995). [5] M. Bardi, Explicit solutions of some linear-quadratic mean field games,, Netw. Heterog. Media., 7 (2012), 243. doi: 10.3934/nhm.2012.7.243. [6] M. Bardi and F. Priuli, LQG Mean-Field Games with ergodic cost,, Proc. 52nd IEEE Conference on Decision and Control, (2013), 2493. doi: 10.1109/CDC.2013.6760255. [7] A. Bhatt and R. Karandikar, Invariant measures and evolution equations for Markov processes,, Ann. Probab., 21 (1993), 2246. doi: 10.1214/aop/1176989019. [8] P. Billingsley, Probability and Measure,, $3^{rd}$ edition, (1995). [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. doi: 10.1081/PDE-100107815. [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. [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. doi: 10.1070/SM2002v193n07ABEH000665. [12] V. Borkar, A topology for Markov controls,, Appl. Math. Optim., 20 (1989), 55. doi: 10.1007/BF01447645. [13] V. Borkar and M. Ghosh, Stochastic differential games: Occupation measure based approach,, J. Optim. Theory Appl., 73 (1992), 359. doi: 10.1007/BF00940187. [14] R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic diferential games,, SIAM J. Control Optim., 43 (2004), 624. doi: 10.1137/S0363012902411556. [15] A. Calderón and J. Rosenblueth, Minimizing Approximate Original Solutions for Commensurate Delayed Controls,, Appl. Math. Lett., 7 (1994), 5. doi: 10.1016/0893-9659(94)90063-9. [16] R. Durrett, Stochastic Calculus: A Practical Introduction,, CRC Press, (1996). [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. doi: 10.1214/aoms/1177729689. [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. doi: 10.1007/s10957-011-9974-4. [19] 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. [20] K. Fan, Minimax theorems,, Proc. N.A.S.U.S.A., 39 (1953), 32. doi: 10.1073/pnas.39.1.42. [21] G. Folland, Real Analysis. Modern Techniques and Their Applications,, $2^{nd}$ edition, (1999). [22] A. Friedman, Stochastic Differential Equations and Applications,, Vol. 1, (1975). [23] M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1962. doi: 10.1137/S0363012996299302. [24] M. Ghosh and A. Bagchi, Stochastic game with average payoff criterion,, Appl. Math. Optim., 38 (1998), 283. doi: 10.1007/s002459900092. [25] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprinted version, (1998). [26] P. Glynn and S. Meyn, A Liapounov bound for solutions of the Poisson equation,, Ann. Prob., 24 (1996), 916. doi: 10.1214/aop/1039639370. [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. doi: 10.1137/040605345. [28] H. Hernández-Hernández, Existence of nash equilibria in discounted nonzero-sum stochastic games with additive structure,, Morfismos, 6 (2002), 43. [29] O. Hernández-Lerma and J. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria,, Springer, (1996). doi: 10.1007/978-1-4612-0729-0. [30] O. Hernández-Lerma and J. Lasserre, Further topics on Discrete-Time Markov Control Processes,, Springer, (1999). doi: 10.1007/978-1-4612-0561-6. [31] 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. [32] F. Klebaner, Introduction to Stochastic Calculus with Applications,, $2^{nd}$ edition, (2005). doi: 10.1142/p386. [33] A. Knapp, Advanced Real Analysis,, $2^{nd}$ edition, (2005). [34] A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis,, Dover Publications, (1999). [35] N. Krylov, Controlled diffusion processes. Applications of Mathematics,, Springer-Verlag, (1980). [36] H. Kushner, Numerical approximations for nonzero-sum stochastic differential games,, SIAM J.Control Optim., 46 (2007), 1942. doi: 10.1137/050647931. [37] H. Küenle, Equilibrium strategies in stochastic games with additive cost and transition structure,, Internat. Game Theory Rev., 1 (1999), 131. doi: 10.1142/S0219198999000098. [38] 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. [39] H. Morimoto and M. Ohashi, On linear stochastic differential games with average cost criterions,, J. Optimization Theory and Applications, 64 (1990), 127. doi: 10.1007/BF00940027. [40] J. Munkres, Topology,, $2^{nd}$ edition, (2000). [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, 1 (1994), 231. [42] 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. [43] P. Pedregal, Optimization, relaxation and Young measures,, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 27. doi: 10.1090/S0273-0979-99-00774-0. [44] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, $4^{th}$ edition, (1995). doi: 10.1007/978-3-662-03185-8. [45] K. Parthasarathy and T. Steerneman, A tool in establishing total variation convergence,, Proceedings of the American Mathematical Society, 95 (1985), 626. doi: 10.1090/S0002-9939-1985-0810175-X. [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. doi: 10.1007/s001860400392. [47] T. Raghavan, S. Tijs and O. Vrieze, On stochastic games with additive reward and transition structure,, J. Optim. Theory Appl., 47 (1985), 451. doi: 10.1007/BF00942191. [48] J. Rosenblueth, Proper relaxation of optimal control problem,, J. Optim. Theory Appl., 74 (1992), 509. doi: 10.1007/BF00940324. [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., (). [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. doi: 10.1007/BF00532612. [51] L. Young, Lectures on the Calculus of Variations and Optimal Control Theory,, W. B. Saunders, (1969). [52] J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). [53] E. Zeidler, Nonlinear Functional Analysis and its Applications II/A,, Springer-Verlag, (1990). doi: 10.1007/978-1-4612-0985-0.

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##### References:
 [1] R. Adams, Sobolev Spaces,, Academic Press, (1975). [2] R. Akella and P. Kumar, Optimal control of production rate in a failure prone manufacturing system,, IEEE Trans. Automatic Control, 31 (1986), 116. doi: 10.1109/TAC.1986.1104206. [3] 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. [4] E. Balder, Lectures on Young Measures,, Cahier de Mathématiques de la Décision, (1995). [5] M. Bardi, Explicit solutions of some linear-quadratic mean field games,, Netw. Heterog. Media., 7 (2012), 243. doi: 10.3934/nhm.2012.7.243. [6] M. Bardi and F. Priuli, LQG Mean-Field Games with ergodic cost,, Proc. 52nd IEEE Conference on Decision and Control, (2013), 2493. doi: 10.1109/CDC.2013.6760255. [7] A. Bhatt and R. Karandikar, Invariant measures and evolution equations for Markov processes,, Ann. Probab., 21 (1993), 2246. doi: 10.1214/aop/1176989019. [8] P. Billingsley, Probability and Measure,, $3^{rd}$ edition, (1995). [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. doi: 10.1081/PDE-100107815. [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. [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. doi: 10.1070/SM2002v193n07ABEH000665. [12] V. Borkar, A topology for Markov controls,, Appl. Math. Optim., 20 (1989), 55. doi: 10.1007/BF01447645. [13] V. Borkar and M. Ghosh, Stochastic differential games: Occupation measure based approach,, J. Optim. Theory Appl., 73 (1992), 359. doi: 10.1007/BF00940187. [14] R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic diferential games,, SIAM J. Control Optim., 43 (2004), 624. doi: 10.1137/S0363012902411556. [15] A. Calderón and J. Rosenblueth, Minimizing Approximate Original Solutions for Commensurate Delayed Controls,, Appl. Math. Lett., 7 (1994), 5. doi: 10.1016/0893-9659(94)90063-9. [16] R. Durrett, Stochastic Calculus: A Practical Introduction,, CRC Press, (1996). [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. doi: 10.1214/aoms/1177729689. [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. doi: 10.1007/s10957-011-9974-4. [19] 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. [20] K. Fan, Minimax theorems,, Proc. N.A.S.U.S.A., 39 (1953), 32. doi: 10.1073/pnas.39.1.42. [21] G. Folland, Real Analysis. Modern Techniques and Their Applications,, $2^{nd}$ edition, (1999). [22] A. Friedman, Stochastic Differential Equations and Applications,, Vol. 1, (1975). [23] M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1962. doi: 10.1137/S0363012996299302. [24] M. Ghosh and A. Bagchi, Stochastic game with average payoff criterion,, Appl. Math. Optim., 38 (1998), 283. doi: 10.1007/s002459900092. [25] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprinted version, (1998). [26] P. Glynn and S. Meyn, A Liapounov bound for solutions of the Poisson equation,, Ann. Prob., 24 (1996), 916. doi: 10.1214/aop/1039639370. [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. doi: 10.1137/040605345. [28] H. Hernández-Hernández, Existence of nash equilibria in discounted nonzero-sum stochastic games with additive structure,, Morfismos, 6 (2002), 43. [29] O. Hernández-Lerma and J. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria,, Springer, (1996). doi: 10.1007/978-1-4612-0729-0. [30] O. Hernández-Lerma and J. Lasserre, Further topics on Discrete-Time Markov Control Processes,, Springer, (1999). doi: 10.1007/978-1-4612-0561-6. [31] 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. [32] F. Klebaner, Introduction to Stochastic Calculus with Applications,, $2^{nd}$ edition, (2005). doi: 10.1142/p386. [33] A. Knapp, Advanced Real Analysis,, $2^{nd}$ edition, (2005). [34] A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis,, Dover Publications, (1999). [35] N. Krylov, Controlled diffusion processes. Applications of Mathematics,, Springer-Verlag, (1980). [36] H. Kushner, Numerical approximations for nonzero-sum stochastic differential games,, SIAM J.Control Optim., 46 (2007), 1942. doi: 10.1137/050647931. [37] H. Küenle, Equilibrium strategies in stochastic games with additive cost and transition structure,, Internat. Game Theory Rev., 1 (1999), 131. doi: 10.1142/S0219198999000098. [38] 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. [39] H. Morimoto and M. Ohashi, On linear stochastic differential games with average cost criterions,, J. Optimization Theory and Applications, 64 (1990), 127. doi: 10.1007/BF00940027. [40] J. Munkres, Topology,, $2^{nd}$ edition, (2000). [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, 1 (1994), 231. [42] 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. [43] P. Pedregal, Optimization, relaxation and Young measures,, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 27. doi: 10.1090/S0273-0979-99-00774-0. [44] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, $4^{th}$ edition, (1995). doi: 10.1007/978-3-662-03185-8. [45] K. Parthasarathy and T. Steerneman, A tool in establishing total variation convergence,, Proceedings of the American Mathematical Society, 95 (1985), 626. doi: 10.1090/S0002-9939-1985-0810175-X. [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. doi: 10.1007/s001860400392. [47] T. Raghavan, S. Tijs and O. Vrieze, On stochastic games with additive reward and transition structure,, J. Optim. Theory Appl., 47 (1985), 451. doi: 10.1007/BF00942191. [48] J. Rosenblueth, Proper relaxation of optimal control problem,, J. Optim. Theory Appl., 74 (1992), 509. doi: 10.1007/BF00940324. [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., (). [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. doi: 10.1007/BF00532612. [51] L. Young, Lectures on the Calculus of Variations and Optimal Control Theory,, W. B. Saunders, (1969). [52] J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). [53] E. Zeidler, Nonlinear Functional Analysis and its Applications II/A,, Springer-Verlag, (1990). doi: 10.1007/978-1-4612-0985-0.
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