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On Tauberian theorem for stationary Nash equilibria

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Abstract

We consider general n-player nonzero-sum dynamic games, which is broader than differential games and could accommodate both discrete and continuous time. Assuming common dynamics, we study the long run average family and discounting average family of the running costs. For each of these game families, we investigate asymptotic properties of its Nash equilibria. We analyze asymptotic Nash equilibria—strategy profiles that are approximately optimal if the planning horizon tends to infinity in long run average games and if the discount tends to zero in discounting games. Moreover, we also assume that this strategy profile is stationary. Under a mild assumption on players’ strategy sets, we prove a uniform Tauberian theorem for stationary asymptotic Nash equilibrium. If a stationary strategy profile is an asymptotic Nash equilibrium and the corresponding Nash value functions converge uniformly for one of the families (when discount goes to zero for discounting games, when planning horizon goes to infinity in long run average games), then for the other family this strategy profile is also an asymptotic Nash equilibrium, and its Nash value functions converge uniformly to the same limit. As an example of application of this theorem, we consider Sorger’ model of competition of two firms.

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References

  1. Artstein, Z., Gaitsgory, V.: The value function of singularly perturbed control systems. Appl. Math. Optim. 41(3), 425–445 (2000). https://doi.org/10.1007/s002459911022

    Article  MathSciNet  MATH  Google Scholar 

  2. Averboukh, Y.V.: Universal Nash equilibrium strategies for differential games. J. Dyn. Control Syst. 21(3), 329–350 (2015). https://doi.org/10.1007/s10883-014-9224-9

    Article  MathSciNet  MATH  Google Scholar 

  3. Bardi, M.: On differential games with long-time-average cost. In: Pourtallier, O., Gaitsgory, V., Bernhard, P. (eds.) Advances in Dynamic Games and Their Applications. Birkhäuser, Boston, pp. 3–18 (2009). https://doi.org/10.1007/978-0-8176-4834-3_1

    Google Scholar 

  4. Barelli, P., Duggan, J.: Extremal choice equilibrium with applications to large games, stochastic games, and endogenous institutions. J. Econ. Theory 155, 95–130 (2015). https://doi.org/10.1016/j.jet.2014.11.010

    Article  MathSciNet  MATH  Google Scholar 

  5. Başar, T., Haurie, A., Zaccour, G.: Nonzero-sum differential games. In: Başar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Berlin (2016). https://doi.org/10.1007/978-3-319-27335-8_5-1

    Chapter  Google Scholar 

  6. Bewley, T., Kohlberg, E.: The asymptotic theory of stochastic games. Math. Oper. Res. 1, 197–208 (1976). https://doi.org/10.1287/moor.1.3.197

    Article  MathSciNet  MATH  Google Scholar 

  7. Cannarsa, P., Quincampoix, M.: Vanishing discount limit and nonexpansive optimal control and differential games. SIAM J. Control Optim. 53(4), 1789–1814 (2015). https://doi.org/10.1137/130945429

    Article  MathSciNet  MATH  Google Scholar 

  8. Escobedo-Trujillo, B.A., Jasso-Fuentes, H., Lopez-Barrientos J.D.: Blackwell–Nash equilibria in zero-sum stochastic differential games In: Hernandez-Hernandez, D., Pardo, J.C., Rivero, V. (eds.) XII Symposium of Probability and Stochastic Processes, pp. 169–193. Birkhauser, Cham (2018). https://doi.org/10.1007/978-3-319-77643-9_5

    Chapter  Google Scholar 

  9. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, New York (1971)

    MATH  Google Scholar 

  10. Gaitsgory, V.: Application of the averaging method for constructing suboptimal solutions of singularly perturbed problems of optimal control. Autom. Remote Control 46, 1081–1088 (1985)

    Google Scholar 

  11. Grüne, L.: On the relation between discounted and average optimal value functions. J. Differ. Equ. 148, 65–99 (1998). https://doi.org/10.1006/jdeq.1998.3451

    Article  MathSciNet  MATH  Google Scholar 

  12. Hardy, G.H.: Divergent Series. Clarendon Press, Oxford (1949)

    MATH  Google Scholar 

  13. He, W., Sun, Y.: Stationary Markov perfect equilibria in discounted stochastic games. J. Econ. Theory 169, 35–61 (2017). https://doi.org/10.1016/j.jet.2017.01.007

    Article  MathSciNet  MATH  Google Scholar 

  14. Jaśkiewicz, A., Nowak, A.S.: Non-zero-sum stochastic games. In: Başar, T., Zaccour, G. (eds.) Handbook of Dynamic Game Theory. Springer, Berlin (2018). https://doi.org/10.1007/978-3-319-27335-8_33-3

    Google Scholar 

  15. Khlopin, D.V.: On asymptotic value for dynamic games with saddle point. In: Bonnet, C., Pasik-Duncan, B., Ozbay, H., Zhang, Q. (eds.) 2015 Proceedings of the Conference on Control and Its Applications, pp. 282–289. SIAM, Philadelphia (2015). https://doi.org/10.1137/1.9781611974072.39

    Chapter  Google Scholar 

  16. Khlopin, D.V.: Uniform Tauberian theorem for differential games. Mat. Teor. Igr Prilozh. 1, 92–120 (2015) (in Russian); in English translate: Automat. Remote Control 77(4), 734–750 (2016) https://doi.org/10.1134/S0005117916040172

    Article  MathSciNet  Google Scholar 

  17. Khlopin, D.V.: On limit of value functions for various densities. In: CEUR Workshop Proceedings, vol. 1987, pp. 328–335 (2017)

  18. Khlopin, D.V.: Value asymptotics in dynamic games on large horizons. arXiv preprint arXiv:1706.08150v2 (2017) (in Russian)

  19. Khlopin, D.V.: On uniform Tauberian theorems for dynamic games. Mat. Sb. 209(1), 127–150 (2018). https://doi.org/10.1070/SM8785

    Article  MathSciNet  Google Scholar 

  20. Khlopin, D.V.: Tauberian theorem for value functions. Dyn. Games Appl. 8, 401 (2018). https://doi.org/10.1007/s13235-017-0227-5

    Article  MathSciNet  MATH  Google Scholar 

  21. Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York (1988)

    Book  Google Scholar 

  22. Lehrer, E., Sorin, S.: A uniform Tauberian theorem in dynamic programming. Math. Oper. Res. 17(2), 303–307 (1992). https://doi.org/10.1287/moor.17.2.303

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, X., Quincampoix, M., Renault, J.: Limit value for optimal control with general means. Discrete Contin. Dyn. Syst. Ser. A 36, 2113–2132 (2016). https://doi.org/10.3934/dcds.2016.36.2113

    Article  MathSciNet  MATH  Google Scholar 

  24. Lions, P., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton–Jacobi Equations, unpublished work (1986)

  25. Marin-Solano, J., Shevkoplyas, E.V.: Non-constant discounting and differential games with random time horizon. Automatica 47(12), 2626–2638 (2011). https://doi.org/10.1016/j.automatica.2011.09.010

    Article  MathSciNet  MATH  Google Scholar 

  26. Mertens, J.F., Neyman, A.: Stochastic games. Int. J. Game Theory 10(2), 53–66 (1981). https://doi.org/10.1007/BF01769259

    Article  MathSciNet  MATH  Google Scholar 

  27. Oliu-Barton, M., Vigeral, G.: A uniform Tauberian theorem in optimal control. In: Cardaliaguet, P., Cressman, R. (eds.) Advances in Dynamic Games. pp. 199–215. Birkhäuser, Boston (2013) https://doi.org/10.1007/978-0-8176-8355-9_10. Erratum: HAL preprint hal:00661833v3, 2016

    MATH  Google Scholar 

  28. Renault, J.: General limit value in dynamic programming. J. Dyn. Games 1(3), 471–484 (2013). https://doi.org/10.3934/jdg.2014.1.471

    Article  MathSciNet  MATH  Google Scholar 

  29. Renault, J., Ziliotto, B.: Hidden stochastic games and limit equilibrium payoffs. arXiv preprint arXiv:1407.3028

  30. Solan, E.: Acceptable strategy profiles in stochastic games. Games Econ. Behav. 108, 523–540 (2018). https://doi.org/10.1016/j.geb.2017.01.011

    Article  MathSciNet  MATH  Google Scholar 

  31. Sorger, G.: Competitive dynamic advertising: a modification of the case game. J. Econ. Dyn. Control 13, 55–80 (1989). https://doi.org/10.1016/0165-1889(89)90011-0

    Article  MathSciNet  MATH  Google Scholar 

  32. Ziliotto, B.: A Tauberian theorem for nonexpansive operators and applications to zero-sum stochastic games. Math. Oper. Res. 41(4), 1522–1534 (2016). https://doi.org/10.1287/moor.2016.0788

    Article  MathSciNet  MATH  Google Scholar 

  33. Ziliotto, B.: General limit value in zero-sum stochastic games. Int. J. Game Theory 45, 353–374 (2016). https://doi.org/10.1007/s00182-015-0509-3

    Article  MathSciNet  MATH  Google Scholar 

  34. Ziliotto, B.: Tauberian theorems for general iterations of operators: applications to zero-sum stochastic games. Games Econ. Behav. 108, 486–503 (2018). https://doi.org/10.1016/j.geb.2018.01.009

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This study was supported by the Russian Science Foundation (Project No. 17-11-01093). I would like to express my gratitude to Ya. Salii for the translation.

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Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Russian Academy of Sciences, 16, S. Kovalevskaja St., Yekaterinburg, Russia, 620990

    D. V. Khlopin

  2. Chair of Applied Mathematics and Mechanics, Institute of Mathematics and Computer Science, Ural Federal University, 4, Turgeneva St., Yekaterinburg, Russia, 620083

    D. V. Khlopin

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Khlopin, D.V. On Tauberian theorem for stationary Nash equilibria. Optim Lett 13, 1855–1870 (2019). https://doi.org/10.1007/s11590-018-1345-8

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