IDEAS home Printed from https://ideas.repec.org/a/spr/dyngam/v9y2019i2d10.1007_s13235-018-0265-7.html
   My bibliography  Save this article

Metastability in Stochastic Replicator Dynamics

Author

Listed:
  • Konstantin Avrachenkov

    (INRIA Sophia Antipolis)

  • Vivek S. Borkar

    (Indian Institute of Technology Bombay)

Abstract

We consider a novel model of stochastic replicator dynamics for potential games that converts to a Langevin equation on a sphere after a change of variables. This is distinct from the models of stochastic replicator dynamics studied earlier. In particular, it is ill-posed due to non-uniqueness of solutions, but is amenable to a natural selection principle that picks a unique solution. The model allows us to make specific statements regarding metastable states such as small noise asymptotics for mean exit times from their domain of attraction, and quasi-stationary measures. We illustrate the general results by specializing them to replicator dynamics on graphs and demonstrate that the numerical experiments support theoretical predictions.

Suggested Citation

  • Konstantin Avrachenkov & Vivek S. Borkar, 2019. "Metastability in Stochastic Replicator Dynamics," Dynamic Games and Applications, Springer, vol. 9(2), pages 366-390, June.
    • Handle: RePEc:spr:dyngam:v:9:y:2019:i:2:d:10.1007_s13235-018-0265-7
    DOI: 10.1007/s13235-018-0265-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13235-018-0265-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s13235-018-0265-7?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384, December.
    2. Fudenberg, Drew & Levine, David, 1998. "Learning in games," European Economic Review, Elsevier, vol. 42(3-5), pages 631-639, May.
    3. Sandholm, William H. & DokumacI, Emin & Lahkar, Ratul, 2008. "The projection dynamic and the replicator dynamic," Games and Economic Behavior, Elsevier, vol. 64(2), pages 666-683, November.
    4. Corradi, Valentina & Sarin, Rajiv, 2008. "Corrigendum to "Continuous approximations of stochastic evolutionary game dynamics" [J. Econ. Theory 94 (2000) 163-191]," Journal of Economic Theory, Elsevier, vol. 140(1), pages 2-4, May.
    5. Valentina Corradi & Rajiv Sarin, "undated". "Continuous Approximations of Stochastic Evolutionary Game Dynamics," ELSE working papers 002, ESRC Centre on Economics Learning and Social Evolution.
    6. Sandholm, William H. & Staudigl, Mathias, 2016. "Large deviations and stochastic stability in the small noise double limit," Theoretical Economics, Econometric Society, vol. 11(1), January.
    7. Mäs, Michael & Nax, Heinrich H., 2016. "A behavioral study of “noise” in coordination games," Journal of Economic Theory, Elsevier, vol. 162(C), pages 195-208.
    8. Borgers, Tilman & Sarin, Rajiv, 1997. "Learning Through Reinforcement and Replicator Dynamics," Journal of Economic Theory, Elsevier, vol. 77(1), pages 1-14, November.
    9. Katsuhiko Aiba, 2015. "Waiting times in evolutionary dynamics with time-decreasing noise," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(2), pages 499-514, May.
    10. Fudenberg, D. & Harris, C., 1992. "Evolutionary dynamics with aggregate shocks," Journal of Economic Theory, Elsevier, vol. 57(2), pages 420-441, August.
    11. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, December.
    12. Sandholm, William H. & Staudigl, Mathias, 2016. "Large Deviations and Stochastic Stability in the Small Noise Double Limit, I: Theory," Center for Mathematical Economics Working Papers 505, Center for Mathematical Economics, Bielefeld University.
    13. Sandholm, William H. & Staudigl, Mathias, 2016. "Large Deviations and Stochastic Stability in the Small Noise Double Limit, II: The Logit Model," Center for Mathematical Economics Working Papers 506, Center for Mathematical Economics, Bielefeld University.
    14. Corradi, Valentina & Sarin, Rajiv, 2000. "Continuous Approximations of Stochastic Evolutionary Game Dynamics," Journal of Economic Theory, Elsevier, vol. 94(2), pages 163-191, October.
    15. Bruce Hajek, 1988. "Cooling Schedules for Optimal Annealing," Mathematics of Operations Research, INFORMS, vol. 13(2), pages 311-329, May.
    16. Cabrales, Antonio, 2000. "Stochastic Replicator Dynamics," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 41(2), pages 451-481, May.
    17. Xing Gao & Weijun Zhong & Shue Mei, 2013. "Stochastic Evolutionary Game Dynamics and Their Selection Mechanisms," Computational Economics, Springer;Society for Computational Economics, vol. 41(2), pages 233-247, February.
    18. Mäs, Michael & Nax, Heinrich H., 2016. "A behavioral study of “noise” in coordination games," LSE Research Online Documents on Economics 65422, London School of Economics and Political Science, LSE Library.
    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. Xinmin Liu & Kangkang Lin & Lei Wang & Hongkun Zhang, 2021. "Stochastic Evolutionary Game Analysis Between Special Committees and CEO: Incentive and Supervision," Dynamic Games and Applications, Springer, vol. 11(3), pages 538-555, September.
    2. Thomas Graham & Maria Kleshnina & Jerzy A. Filar, 2023. "Where Do Mistakes Lead? A Survey of Games with Incompetent Players," Dynamic Games and Applications, Springer, vol. 13(1), pages 231-264, March.
    3. Sylvain Sorin, 2023. "Continuous Time Learning Algorithms in Optimization and Game Theory," Dynamic Games and Applications, Springer, vol. 13(1), pages 3-24, March.

    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. Jonathan Newton, 2018. "Evolutionary Game Theory: A Renaissance," Games, MDPI, vol. 9(2), pages 1-67, May.
    2. Williams, Noah, 2022. "Learning and equilibrium transitions: Stochastic stability in discounted stochastic fictitious play," Journal of Economic Dynamics and Control, Elsevier, vol. 145(C).
    3. Sawa, Ryoji & Wu, Jiabin, 2018. "Prospect dynamics and loss dominance," Games and Economic Behavior, Elsevier, vol. 112(C), pages 98-124.
    4. Sawa, Ryoji & Wu, Jiabin, 2018. "Reference-dependent preferences, super-dominance and stochastic stability," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 96-104.
    5. Weibull, Jörgen W., 1997. "What have we learned from Evolutionary Game Theory so far?," Working Paper Series 487, Research Institute of Industrial Economics, revised 26 Oct 1998.
    6. Battalio,R. & Samuelson,L. & Huyck,J. van, 1998. "Risk dominance, payoff dominance and probabilistic choice learning," Working papers 2, Wisconsin Madison - Social Systems.
    7. He, Simin & Wu, Jiabin, 2020. "Compromise and coordination: An experimental study," Games and Economic Behavior, Elsevier, vol. 119(C), pages 216-233.
    8. William H. Sandholm & Mathias Staudigl, 2018. "Sample Path Large Deviations for Stochastic Evolutionary Game Dynamics," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1348-1377, November.
    9. 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.
    10. Sandholm,W.H., 1999. "Markov evolution with inexact information," Working papers 15, Wisconsin Madison - Social Systems.
    11. Izquierdo, Luis R. & Izquierdo, Segismundo S. & Sandholm, William H., 2019. "An introduction to ABED: Agent-based simulation of evolutionary game dynamics," Games and Economic Behavior, Elsevier, vol. 118(C), pages 434-462.
    12. Sandholm, William H., 2003. "Evolution and equilibrium under inexact information," Games and Economic Behavior, Elsevier, vol. 44(2), pages 343-378, August.
    13. Ianni, A., 2002. "Reinforcement learning and the power law of practice: some analytical results," Discussion Paper Series In Economics And Econometrics 203, Economics Division, School of Social Sciences, University of Southampton.
    14. Tsakas, Elias & Voorneveld, Mark, 2009. "The target projection dynamic," Games and Economic Behavior, Elsevier, vol. 67(2), pages 708-719, November.
    15. Demichelis, Stefano & Ritzberger, Klaus, 2003. "From evolutionary to strategic stability," Journal of Economic Theory, Elsevier, vol. 113(1), pages 51-75, November.
    16. Weibull, Jorgen W., 1998. "Evolution, rationality and equilibrium in games," European Economic Review, Elsevier, vol. 42(3-5), pages 641-649, May.
    17. Lim, Wooyoung & Neary, Philip R., 2016. "An experimental investigation of stochastic adjustment dynamics," Games and Economic Behavior, Elsevier, vol. 100(C), pages 208-219.
    18. Mertikopoulos, Panayotis & Sandholm, William H., 2018. "Riemannian game dynamics," Journal of Economic Theory, Elsevier, vol. 177(C), pages 315-364.
    19. Jordi Brandts & Antonio Cabrales & Gary Charness, 2003. "Forward induction and the excess capacity puzzle: An experimental investigation," Economics Working Papers 703, Department of Economics and Business, Universitat Pompeu Fabra.
    20. Arigapudi, Srinivas, 2024. "Transitions between equilibria in Bilingual Games under Probit Choice," Journal of Mathematical Economics, Elsevier, vol. 111(C).

    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:spr:dyngam:v:9:y:2019:i:2:d:10.1007_s13235-018-0265-7. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.