IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-00360767.html
   My bibliography  Save this paper

Time Average Replicator and Best Reply Dynamics

Author

Listed:
  • Josef Hofbauer

    (Department of Mathematics, University of Vienna - Universität Wien = University of Vienna)

  • Sylvain Sorin

    (C&O - Equipe combinatoire et optimisation - IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique, X-DEP-ECO - Département d'Économie de l'École Polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris)

  • Yannick Viossat

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

Abstract

Using an explicit representation in terms of the logit map we show, in a unilateral framework, that the time average of the replicator dynamics is a perturbed solution of the best reply dynamics.

Suggested Citation

  • Josef Hofbauer & Sylvain Sorin & Yannick Viossat, 2009. "Time Average Replicator and Best Reply Dynamics," Post-Print hal-00360767, HAL.
    • Handle: RePEc:hal:journl:hal-00360767
    DOI: 10.1287/moor.1080.0359
    Note: View the original document on HAL open archive server: https://hal.science/hal-00360767
    as

    Download full text from publisher

    File URL: https://hal.science/hal-00360767/document
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.1080.0359?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Viossat, Yannick, 2007. "The replicator dynamics does not lead to correlated equilibria," Games and Economic Behavior, Elsevier, vol. 59(2), pages 397-407, May.
    2. Foster, Dean P. & Vohra, Rakesh, 1999. "Regret in the On-Line Decision Problem," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 7-35, October.
    3. Fudenberg, Drew & Levine, David K., 1995. "Consistency and cautious fictitious play," Journal of Economic Dynamics and Control, Elsevier, vol. 19(5-7), pages 1065-1089.
    4. Gaunersdorfer Andrea & Hofbauer Josef, 1995. "Fictitious Play, Shapley Polygons, and the Replicator Equation," Games and Economic Behavior, Elsevier, vol. 11(2), pages 279-303, November.
    5. Ed Hopkins, 2002. "Two Competing Models of How People Learn in Games," Econometrica, Econometric Society, vol. 70(6), pages 2141-2166, November.
    6. Sergiu Hart & Andreu Mas-Colell, 2013. "A Simple Adaptive Procedure Leading To Correlated Equilibrium," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 2, pages 17-46, World Scientific Publishing Co. Pte. Ltd..
    7. Gilboa, Itzhak & Matsui, Akihiko, 1991. "Social Stability and Equilibrium," Econometrica, Econometric Society, vol. 59(3), pages 859-867, May.
    8. Michel Benaim & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions II: Applications," Levine's Bibliography 784828000000000098, UCLA Department of Economics.
    9. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2003. "Stochastic Approximations and Differential Inclusions," Working Papers hal-00242990, HAL.
    10. repec:dau:papers:123456789/1061 is not listed on IDEAS
    11. Freund, Yoav & Schapire, Robert E., 1999. "Adaptive Game Playing Using Multiplicative Weights," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 79-103, October.
    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. Viossat, Yannick & Zapechelnyuk, Andriy, 2013. "No-regret dynamics and fictitious play," Journal of Economic Theory, Elsevier, vol. 148(2), pages 825-842.
    2. Bervoets, Sebastian & Faure, Mathieu, 2020. "Convergence in games with continua of equilibria," Journal of Mathematical Economics, Elsevier, vol. 90(C), pages 25-30.
    3. repec:hal:wpaper:hal-00713871 is not listed on IDEAS
    4. van Strien, Sebastian & Sparrow, Colin, 2011. "Fictitious play in 3x3 games: Chaos and dithering behaviour," Games and Economic Behavior, Elsevier, vol. 73(1), pages 262-286, September.
    5. Norman, Thomas W.L., 2020. "Market selection with an endogenous state," Journal of Mathematical Economics, Elsevier, vol. 91(C), pages 51-59.
    6. Mertikopoulos, Panayotis & Sandholm, William H., 2018. "Riemannian game dynamics," Journal of Economic Theory, Elsevier, vol. 177(C), pages 315-364.
    7. Yannick Viossat, 2012. "Game Dynamics and Nash Equilibria," Working Papers hal-00756096, HAL.
    8. Pierre Coucheney & Bruno Gaujal & Panayotis Mertikopoulos, 2015. "Penalty-Regulated Dynamics and Robust Learning Procedures in Games," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 611-633, March.
    9. Russell Golman, 2011. "Why learning doesn’t add up: equilibrium selection with a composition of learning rules," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(4), pages 719-733, November.
    10. Jonathan Newton, 2018. "Evolutionary Game Theory: A Renaissance," Games, MDPI, vol. 9(2), pages 1-67, May.
    11. Sandholm, William H., 2015. "Population Games and Deterministic Evolutionary Dynamics," Handbook of Game Theory with Economic Applications,, Elsevier.
    12. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2012. "Perturbations of Set-Valued Dynamical Systems, with Applications to Game Theory," Dynamic Games and Applications, Springer, vol. 2(2), pages 195-205, June.
    13. Lahkar, Ratul & Seymour, Robert M., 2013. "Reinforcement learning in population games," Games and Economic Behavior, Elsevier, vol. 80(C), pages 10-38.
    14. Sofia B. S. D. Castro, 2018. "Learning by replicator and best-response: the importance of being indifferent," Journal of Evolutionary Economics, Springer, vol. 28(4), pages 985-999, September.
    15. Yannick Viossat, 2014. "Game Dynamics and Nash Equilibria," Post-Print hal-00756096, HAL.
    16. Sylvain Sorin, 2023. "Continuous Time Learning Algorithms in Optimization and Game Theory," Dynamic Games and Applications, Springer, vol. 13(1), pages 3-24, March.
    17. Saeed Hadikhanloo & Rida Laraki & Panayotis Mertikopoulos & Sylvain Sorin, 2022. "Learning in nonatomic games, part Ⅰ: Finite action spaces and population games," Post-Print hal-03767995, HAL.
    18. Panayotis Mertikopoulos & William H. Sandholm, 2016. "Learning in Games via Reinforcement and Regularization," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1297-1324, November.
    19. Laraki, Rida & Mertikopoulos, Panayotis, 2013. "Higher order game dynamics," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2666-2695.
    20. Liliana Garrido-da-Silva & Sofia B. S. D. Castro, 2020. "Cyclic dominance in a two-person rock–scissors–paper game," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(3), pages 885-912, September.
    21. Michel Benaïm & Mathieu Faure, 2013. "Consistency of Vanishingly Smooth Fictitious Play," Mathematics of Operations Research, INFORMS, vol. 38(3), pages 437-450, August.

    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. Josef Hofbauer & Sylvain Sorin & Yannick Viossat, 2009. "Time Average Replicator and Best-Reply Dynamics," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 263-269, May.
    2. Andriy Zapechelnyuk, 2009. "Limit Behavior of No-regret Dynamics," Discussion Papers 21, Kyiv School of Economics.
    3. Benaïm, Michel & Hofbauer, Josef & Hopkins, Ed, 2009. "Learning in games with unstable equilibria," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1694-1709, July.
    4. Viossat, Yannick & Zapechelnyuk, Andriy, 2013. "No-regret dynamics and fictitious play," Journal of Economic Theory, Elsevier, vol. 148(2), pages 825-842.
    5. repec:hal:wpaper:hal-00713871 is not listed on IDEAS
    6. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions; Part II: Applications," Working Papers hal-00242974, HAL.
    7. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2006. "Stochastic Approximations and Differential Inclusions, Part II: Applications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 673-695, November.
    8. Panayotis Mertikopoulos & William H. Sandholm, 2016. "Learning in Games via Reinforcement and Regularization," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1297-1324, November.
    9. Karl Schlag & Andriy Zapechelnyuk, 2009. "Decision Making in Uncertain and Changing Environments," Discussion Papers 19, Kyiv School of Economics.
    10. Cason, Timothy N. & Friedman, Daniel & Hopkins, Ed, 2010. "Testing the TASP: An experimental investigation of learning in games with unstable equilibria," Journal of Economic Theory, Elsevier, vol. 145(6), pages 2309-2331, November.
    11. Emerson Melo, 2021. "Learning in Random Utility Models Via Online Decision Problems," Papers 2112.10993, arXiv.org, revised Aug 2022.
    12. Mannor, Shie & Shimkin, Nahum, 2008. "Regret minimization in repeated matrix games with variable stage duration," Games and Economic Behavior, Elsevier, vol. 63(1), pages 227-258, May.
    13. Hofbauer, Josef & Sandholm, William H., 2009. "Stable games and their dynamics," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1665-1693.4, July.
    14. Schlag, Karl H. & Zapechelnyuk, Andriy, 2017. "Dynamic benchmark targeting," Journal of Economic Theory, Elsevier, vol. 169(C), pages 145-169.
    15. Schlag, Karl & Zapechelnyuk, Andriy, 2012. "On the impossibility of achieving no regrets in repeated games," Journal of Economic Behavior & Organization, Elsevier, vol. 81(1), pages 153-158.
    16. Sergiu Hart & Andreu Mas-Colell, 2013. "A General Class Of Adaptive Strategies," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 3, pages 47-76, World Scientific Publishing Co. Pte. Ltd..
    17. Sandholm, William H., 2015. "Population Games and Deterministic Evolutionary Dynamics," Handbook of Game Theory with Economic Applications,, Elsevier.
    18. Karl Schlag & Andriy Zapechelnyuk, 2010. "On the Impossibility of Regret Minimization in Repeated Games," Working Papers 676, Queen Mary University of London, School of Economics and Finance.
    19. Emerson Melo, 2021. "Learning In Random Utility Models Via Online Decision Problems," CAEPR Working Papers 2022-003 Classification-D, Center for Applied Economics and Policy Research, Department of Economics, Indiana University Bloomington.
    20. Fudenberg, Drew & Levine, David K., 1999. "Conditional Universal Consistency," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 104-130, October.
    21. Viossat, Yannick, 2008. "Evolutionary dynamics may eliminate all strategies used in correlated equilibrium," Mathematical Social Sciences, Elsevier, vol. 56(1), pages 27-43, July.

    More about this item

    Keywords

    replicator dynamics; best reply dynamics; logit map; perturbed differential inclusion; internally chain transitive set; attractor;
    All these keywords.

    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:hal:journl:hal-00360767. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.