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Tauberian Theorem for Value Functions

Author

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  • Dmitry Khlopin

    (Ural Branch, Russian Academy of Sciences
    Ural Federal University)

Abstract

For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long-run average cost as the time horizon tends to infinity and the limit of value functions of $$\lambda $$ λ -discounted games as the discount tends to zero. We prove that the Dynamic Programming Principle for value functions directly leads to the Tauberian theorem—that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and superoptimality principles (the Dynamic Programming Principle). With their aid, we obtain certain inequalities on asymptotics of sub- and supersolutions, which lead to the Tauberian theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point; a very simple stochastic game model is also considered.

Suggested Citation

  • Dmitry Khlopin, 2018. "Tauberian Theorem for Value Functions," Dynamic Games and Applications, Springer, vol. 8(2), pages 401-422, June.
    • Handle: RePEc:spr:dyngam:v:8:y:2018:i:2:d:10.1007_s13235-017-0227-5
    DOI: 10.1007/s13235-017-0227-5
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    References listed on IDEAS

    as
    1. Sylvain Sorin, 2011. "Zero-Sum Repeated Games: Recent Advances and New Links with Differential Games," Dynamic Games and Applications, Springer, vol. 1(1), pages 172-207, March.
    2. Monderer, Dov & Sorin, Sylvain, 1993. "Asymptotic Properties in Dynamic Programming," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(1), pages 1-11.
    3. Truman Bewley & Elon Kohlberg, 1976. "The Asymptotic Theory of Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 197-208, August.
    4. Bruno Ziliotto, 2016. "A Tauberian Theorem for Nonexpansive Operators and Applications to Zero-Sum Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1522-1534, November.
    5. Ehud Lehrer & Sylvain Sorin, 1992. "A Uniform Tauberian Theorem in Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 17(2), pages 303-307, May.
    6. Xiaoxi Li & Xavier Venel, 2016. "Recursive games: Uniform value, Tauberian theorem and the Mertens conjecture " M axmin = lim v n = lim v λ "," Post-Print hal-01302553, HAL.
    7. R. Buckdahn & P. Cardaliaguet & M. Quincampoix, 2011. "Some Recent Aspects of Differential Game Theory," Dynamic Games and Applications, Springer, vol. 1(1), pages 74-114, March.
    8. Guillaume Vigeral, 2013. "A Zero-Sum Stochastic Game with Compact Action Sets and no Asymptotic Value," Dynamic Games and Applications, Springer, vol. 3(2), pages 172-186, June.
    9. Bruno Ziliotto, 2016. "General limit value in zero-sum stochastic games," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(1), pages 353-374, March.
    10. Xiaoxi Li & Xavier Venel, 2016. "Recursive games: Uniform value, Tauberian theorem and the Mertens conjecture " M axmin = lim v n = lim v λ "," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01302553, HAL.
    11. repec:dau:papers:123456789/6046 is not listed on IDEAS
    12. repec:dau:papers:123456789/10880 is not listed on IDEAS
    13. Xiaoxi Li & Xavier Venel, 2016. "Recursive games: Uniform value, Tauberian theorem and the Mertens conjecture " M axmin = lim v n = lim v λ "," PSE-Ecole d'économie de Paris (Postprint) hal-01302553, HAL.
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    Cited by:

    1. Antoine Hochart, 2021. "Unique Ergodicity of Deterministic Zero-Sum Differential Games," Dynamic Games and Applications, Springer, vol. 11(1), pages 109-136, March.

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