IDEAS home Printed from https://ideas.repec.org/p/tse/wpaper/31602.html
   My bibliography  Save this paper

How Fast Do Equilibrium Payo Sets Converge in Repeated Games?

Author

Listed:
  • Hörner, Johannes
  • Takahashi, Satoru

Abstract

We provide tight bounds on the rate of convergence of the equilibrium payoff sets for repeated games under both perfect and imperfect public monitoring. The distance between the equilibrium payoff set and its limit vanishes at rate (1−δ)1/2 under perfect monitoring, and at rate (1−δ)1/4 under imperfect monitoring. For strictly individually rational payoff vectors, these rates improve to 0 (i.e., all strictly individually rational payoff vectors are exactly achieved as equilibrium payoffs for δ high enough) and (1−δ)1/2, respectively.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Hörner, Johannes & Takahashi, Satoru, 2017. "How Fast Do Equilibrium Payo Sets Converge in Repeated Games?," TSE Working Papers 17-792, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:31602
    as

    Download full text from publisher

    File URL: https://www.tse-fr.eu/sites/default/files/TSE/documents/doc/by/horner/rate_of_convergence03.pdf
    File Function: Full text
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Yu Awaya & Vijay Krishna, 2016. "On Communication and Collusion," American Economic Review, American Economic Association, vol. 106(2), pages 285-315, February.
    2. Sugaya, Takuo & Wolitzky, Alexander, 2017. "Bounding equilibrium payoffs in repeated games with private monitoring," Theoretical Economics, Econometric Society, vol. 12(2), May.
    3. Michihiro Kandori & Ichiro Obara, 2006. "Efficiency in Repeated Games Revisited: The Role of Private Strategies," Econometrica, Econometric Society, vol. 74(2), pages 499-519, March.
    4. Drew Fudenberg & David K. Levine & Satoru Takahashi, 2008. "Perfect public equilibrium when players are patient," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 16, pages 345-367, World Scientific Publishing Co. Pte. Ltd..
    5. Drew Fudenberg & David K. Levine, 2008. "Efficiency and Observability with Long-Run and Short-Run Players," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 13, pages 275-307, World Scientific Publishing Co. Pte. Ltd..
    6. Drew Fudenberg & David Levine & Eric Maskin, 2008. "The Folk Theorem With Imperfect Public Information," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 12, pages 231-273, World Scientific Publishing Co. Pte. Ltd..
    7. Jean-FranÚois Mertens, 1998. "The speed of convergence in repeated games with incomplete information on one side," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(3), pages 343-357.
    8. Yuliy Sannikov, 2007. "Games with Imperfectly Observable Actions in Continuous Time," Econometrica, Econometric Society, vol. 75(5), pages 1285-1329, September.
    9. Drew Fudenberg & Eric Maskin, 2008. "The Folk Theorem In Repeated Games With Discounting Or With Incomplete Information," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 11, pages 209-230, World Scientific Publishing Co. Pte. Ltd..
    10. Abreu, Dilip & Milgrom, Paul & Pearce, David, 1991. "Information and Timing in Repeated Partnerships," Econometrica, Econometric Society, vol. 59(6), pages 1713-1733, November.
    11. Tomala, Tristan, 2009. "Perfect communication equilibria in repeated games with imperfect monitoring," Games and Economic Behavior, Elsevier, vol. 67(2), pages 682-694, November.
    12. Yuliy Sannikov & Andrzej Skrzypacz, 2010. "The Role of Information in Repeated Games With Frequent Actions," Econometrica, Econometric Society, vol. 78(3), pages 847-882, May.
    13. Truman Bewley & Elon Kohlberg, 1976. "The Asymptotic Theory of Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 197-208, August.
    14. Fudenberg, D. & Maskin, E., 1990. "Nash and perfect equilibria of discounted repeated games," Journal of Economic Theory, Elsevier, vol. 51(1), pages 194-206, June.
    15. Fudenberg, Drew & Maskin, Eric, 1991. "On the dispensability of public randomization in discounted repeated games," Journal of Economic Theory, Elsevier, vol. 53(2), pages 428-438, April.
    16. Ortigueira, Salvador & Santos, Manuel S, 1997. "On the Speed of Convergence in Endogenous Growth Models," American Economic Review, American Economic Association, vol. 87(3), pages 383-399, June.
    17. Chen, Xiaohong & White, Halbert, 1996. "Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications," Econometric Theory, Cambridge University Press, vol. 12(2), pages 284-304, June.
    18. Mailath, George J. & Obara, Ichiro & Sekiguchi, Tadashi, 2002. "The Maximum Efficient Equilibrium Payoff in the Repeated Prisoners' Dilemma," Games and Economic Behavior, Elsevier, vol. 40(1), pages 99-122, July.
    19. Sorin, Sylvain, 1992. "Repeated games with complete information," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 4, pages 71-107, Elsevier.
    20. Michihiro Kandori, 1992. "The Use of Information in Repeated Games with Imperfect Monitoring," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 59(3), pages 581-593.
    21. Olivier Compte, 1998. "Communication in Repeated Games with Imperfect Private Monitoring," Econometrica, Econometric Society, vol. 66(3), pages 597-626, May.
    22. Abreu, Dilip & Pearce, David & Stacchetti, Ennio, 1990. "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring," Econometrica, Econometric Society, vol. 58(5), pages 1041-1063, September.
    23. Thomas, J. P., 1995. "Subgame-perfect attainment of minimax punishments in discounted two-person games," Economics Letters, Elsevier, vol. 47(1), pages 1-4, January.
    24. Johannes Hörner & Takuo Sugaya & Satoru Takahashi & Nicolas Vieille, 2011. "Recursive Methods in Discounted Stochastic Games: An Algorithm for δ→ 1 and a Folk Theorem," Econometrica, Econometric Society, vol. 79(4), pages 1277-1318, July.
    25. Gossner, Olivier, 1995. "The Folk Theorem for Finitely Repeated Games with Mixed Strategies," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(1), pages 95-107.
    26. repec:dau:papers:123456789/6102 is not listed on IDEAS
    27. Stahl, Dale II, 1991. "The graph of Prisoners' Dilemma supergame payoffs as a function of the discount factor," Games and Economic Behavior, Elsevier, vol. 3(3), pages 368-384, August.
    28. Benoit, Jean-Pierre & Krishna, Vijay, 1985. "Finitely Repeated Games," Econometrica, Econometric Society, vol. 53(4), pages 905-922, July.
    29. Eitan Altman & Ofer Zeitouni, 1994. "Rate of Convergence of Empirical Measures and Costs in Controlled Markov Chains and Transient Optimality," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 955-974, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ani Dasgupta & Sambuddha Ghosh, 2017. "Repeated Games Without Public Randomization: A Constructive Approach," Boston University - Department of Economics - Working Papers Series WP2017-011, Boston University - Department of Economics, revised Feb 2019.
    2. Meng, Delong, 2021. "On the value of repetition for communication games," Games and Economic Behavior, Elsevier, vol. 127(C), pages 227-246.
    3. Matan Harel & Elchanan Mossel & Philipp Strack & Omer Tamuz, 2021. "Rational Groupthink," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 136(1), pages 621-668.
      • Matan Harel & Elchanan Mossel & Philipp Strack & Omer Tamuz, 2014. "Rational Groupthink," Papers 1412.7172, arXiv.org, revised Jun 2020.
    4. Joyee Deb & Takuo Sugaya & Alexander Wolitzky, 2020. "The Folk Theorem in Repeated Games With Anonymous Random Matching," Econometrica, Econometric Society, vol. 88(3), pages 917-964, May.
    5. Dasgupta, Ani & Ghosh, Sambuddha, 2022. "Self-accessibility and repeated games with asymmetric discounting," Journal of Economic Theory, Elsevier, vol. 200(C).
    6. Mira Frick & Ryota Iijima & Yuhta Ishii, 2023. "Monitoring with Rich Data," Papers 2312.16789, arXiv.org, revised Jul 2024.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Laclau, Marie & Tomala, Tristan, 2017. "Repeated games with public deterministic monitoring," Journal of Economic Theory, Elsevier, vol. 169(C), pages 400-424.
    2. Osório António M., 2012. "A Folk Theorem for Games when Frequent Monitoring Decreases Noise," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 12(1), pages 1-27, April.
    3. Fudenberg, Drew & Ishii, Yuhta & Kominers, Scott Duke, 2014. "Delayed-response strategies in repeated games with observation lags," Journal of Economic Theory, Elsevier, vol. 150(C), pages 487-514.
    4. Kimmo Berg & Gijs Schoenmakers, 2017. "Construction of Subgame-Perfect Mixed-Strategy Equilibria in Repeated Games," Games, MDPI, vol. 8(4), pages 1-14, November.
    5. Contou-Carrère, Pauline & Tomala, Tristan, 2011. "Finitely repeated games with semi-standard monitoring," Journal of Mathematical Economics, Elsevier, vol. 47(1), pages 14-21, January.
    6. Johannes Horner & Satoru Takahashi & Nicolas Vieille, 2012. "On the Limit Equilibrium Payoff Set in Repeated and Stochastic Games," Cowles Foundation Discussion Papers 1848, Cowles Foundation for Research in Economics, Yale University.
    7. Dasgupta, Ani & Ghosh, Sambuddha, 2022. "Self-accessibility and repeated games with asymmetric discounting," Journal of Economic Theory, Elsevier, vol. 200(C).
    8. Jérôme Renault & Tristan Tomala, 2011. "General Properties of Long-Run Supergames," Dynamic Games and Applications, Springer, vol. 1(2), pages 319-350, June.
    9. Daehyun Kim & Ichiro Obara, 2023. "Asymptotic Value of Monitoring Structures in Stochastic Games," Papers 2308.09211, arXiv.org, revised Jul 2024.
    10. Aramendia, Miguel & Wen, Quan, 2020. "Myopic perception in repeated games," Games and Economic Behavior, Elsevier, vol. 119(C), pages 1-14.
    11. Hörner, Johannes & Takahashi, Satoru & Vieille, Nicolas, 2014. "On the limit perfect public equilibrium payoff set in repeated and stochastic games," Games and Economic Behavior, Elsevier, vol. 85(C), pages 70-83.
    12. Ashkenazi-Golan, Galit & Lehrer, Ehud, 2019. "What you get is what you see: Cooperation in repeated games with observable payoffs," Journal of Economic Theory, Elsevier, vol. 181(C), pages 197-237.
    13. Kobayashi, Hajime & Ohta, Katsunori, 2012. "Optimal collusion under imperfect monitoring in multimarket contact," Games and Economic Behavior, Elsevier, vol. 76(2), pages 636-647.
    14. Abito, Jose Miguel & Chen, Cuicui, 2023. "A partial identification framework for dynamic games," International Journal of Industrial Organization, Elsevier, vol. 87(C).
    15. Du, Chuang, 2012. "Solving payoff sets of perfect public equilibria: an example," MPRA Paper 38622, University Library of Munich, Germany.
    16. Osório-Costa, António M., 2009. "Efficiency Gains in Repeated Games at Random Moments in Time," MPRA Paper 13105, University Library of Munich, Germany.
    17. Michihiro Kandori, 2011. "Weakly Belief‐Free Equilibria in Repeated Games With Private Monitoring," Econometrica, Econometric Society, vol. 79(3), pages 877-892, May.
    18. Drew Fudenberg & David K. Levine & Satoru Takahashi, 2008. "Perfect public equilibrium when players are patient," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 16, pages 345-367, World Scientific Publishing Co. Pte. Ltd..
    19. Escobar, Juan F. & Llanes, Gastón, 2018. "Cooperation dynamics in repeated games of adverse selection," Journal of Economic Theory, Elsevier, vol. 176(C), pages 408-443.
    20. McLean, Richard & Obara, Ichiro & Postlewaite, Andrew, 2014. "Robustness of public equilibria in repeated games with private monitoring," Journal of Economic Theory, Elsevier, vol. 153(C), pages 191-212.

    More about this item

    Keywords

    Repeated games; rates of convergence;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:tse:wpaper:31602. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/tsetofr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.