IDEAS home Printed from https://ideas.repec.org/a/spr/dyngam/v7y2017i1d10.1007_s13235-015-0164-0.html
   My bibliography  Save this article

On the Evolution of Continuous Types Under Replicator and Gradient Dynamics: Two Examples

Author

Listed:
  • Jean Paul Rabanal

    (Pettengill Hall)

Abstract

This paper illustrates techniques for assessing the dynamic stability of games where a continuum of types might be present by re-analyzing two models under incomplete information, the Lohman et al. (Unpublished manuscript, 2001) public goods game and the Kopel et al. (J Econ Dyn Control 48:394–409, 2014) Cournot duopoly game. The evolution of continuous types follows either replicator dynamics Oechssler and Riedel (Econ Theory 17:141–162, 2001; J Econ Theory 107:223–252 2002) or gradient dynamics Friedman and Ostrov (J Math Econ 46:691–707, 2010; J Econ Theory 148:743–777, 2013). The techniques rely on a system of partial differential equations. Numerical solutions obtained through replicator and gradient dynamics highlight the differences and the similarities that arise under both approaches. In the public goods game, the dynamic system affects the stationary distribution of types while in the Cournot duopoly model, the types evolve to a single mass point regardless of the dynamics used. Lastly, these techniques allow us to endogenize the distribution of player types.

Suggested Citation

  • Jean Paul Rabanal, 2017. "On the Evolution of Continuous Types Under Replicator and Gradient Dynamics: Two Examples," Dynamic Games and Applications, Springer, vol. 7(1), pages 76-92, March.
    • Handle: RePEc:spr:dyngam:v:7:y:2017:i:1:d:10.1007_s13235-015-0164-0
    DOI: 10.1007/s13235-015-0164-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13235-015-0164-0
    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/s13235-015-0164-0?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. Perkins, S. & Leslie, D.S., 2014. "Stochastic fictitious play with continuous action sets," Journal of Economic Theory, Elsevier, vol. 152(C), pages 179-213.
    2. Oechssler, Jorg & Riedel, Frank, 2002. "On the Dynamic Foundation of Evolutionary Stability in Continuous Models," Journal of Economic Theory, Elsevier, vol. 107(2), pages 223-252, December.
    3. Hofbauer, Josef & Weibull, Jorgen W., 1996. "Evolutionary Selection against Dominated Strategies," Journal of Economic Theory, Elsevier, vol. 71(2), pages 558-573, November.
    4. Friedman, Daniel & Singh, Nirvikar, 2009. "Equilibrium vengeance," Games and Economic Behavior, Elsevier, vol. 66(2), pages 813-829, July.
    5. Ritzberger, Klaus & Weibull, Jorgen W, 1995. "Evolutionary Selection in Normal-Form Games," Econometrica, Econometric Society, vol. 63(6), pages 1371-1399, November.
    6. Kopel, Michael & Lamantia, Fabio & Szidarovszky, Ferenc, 2014. "Evolutionary competition in a mixed market with socially concerned firms," Journal of Economic Dynamics and Control, Elsevier, vol. 48(C), pages 394-409.
    7. Hofbauer, Josef & Oechssler, Jörg & Riedel, Frank, 2009. "Brown-von Neumann-Nash dynamics: The continuous strategy case," Games and Economic Behavior, Elsevier, vol. 65(2), pages 406-429, March.
    8. Ok, Efe A. & Vega-Redondo, Fernando, 2001. "On the Evolution of Individualistic Preferences: An Incomplete Information Scenario," Journal of Economic Theory, Elsevier, vol. 97(2), pages 231-254, April.
    9. Cheung, Man-Wah, 2014. "Pairwise comparison dynamics for games with continuous strategy space," Journal of Economic Theory, Elsevier, vol. 153(C), pages 344-375.
    10. Kopel, Michael & Brand, Björn, 2012. "Socially responsible firms and endogenous choice of strategic incentives," Economic Modelling, Elsevier, vol. 29(3), pages 982-989.
    11. Ely, Jeffrey C. & Sandholm, William H., 2005. "Evolution in Bayesian games I: Theory," Games and Economic Behavior, Elsevier, vol. 53(1), pages 83-109, October.
    12. repec:hhs:iuiwop:487 is not listed on IDEAS
    13. Friedman, Daniel & Ostrov, Daniel N., 2010. "Gradient dynamics in population games: Some basic results," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 691-707, September.
    14. Jorgen W. Weibull, 1997. "Evolutionary Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262731215, December.
    15. Jean Rabanal & Daniel Friedman, 2014. "Incomplete Information, Dynamic Stability and the Evolution of Preferences: Two Examples," Dynamic Games and Applications, Springer, vol. 4(4), pages 448-467, December.
    16. Friedman, Daniel & Ostrov, Daniel N., 2013. "Evolutionary dynamics over continuous action spaces for population games that arise from symmetric two-player games," Journal of Economic Theory, Elsevier, vol. 148(2), pages 743-777.
    17. Lahkar, Ratul & Riedel, Frank, 2015. "The logit dynamic for games with continuous strategy sets," Games and Economic Behavior, Elsevier, vol. 91(C), pages 268-282.
    18. Aviad Heifetz & Chris Shannon & Yossi Spiegel, 2007. "The Dynamic Evolution of Preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 32(2), pages 251-286, August.
    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. Domenico De Giovanni & Fabio Lamantia, 2017. "Evolutionary dynamics of a duopoly game with strategic delegation and isoelastic demand," Journal of Evolutionary Economics, Springer, vol. 27(5), pages 877-903, November.
    2. Kopel, Michael & Lamantia, Fabio, 2018. "The persistence of social strategies under increasing competitive pressure," Journal of Economic Dynamics and Control, Elsevier, vol. 91(C), pages 71-83.
    3. Rabanal, Jean Paul & Lee, Dongwook, 2017. "On the dynamic stability of a price dispersion model using gradient dynamics," Journal of Economic Dynamics and Control, Elsevier, vol. 84(C), pages 32-42.
    4. Philip J. Grossman & Youngseok Park & Jean Paul Rabanal & Olga A. Rud, 2019. "Gender differences in an endogenous timing conflict game," Working Papers 141, Peruvian Economic Association.
    5. Fulei Shi & Chuansheng Wang & Cuiyou Yao, 2022. "A New Evolutionary Game Analysis for Industrial Pollution Management Considering the Central Government’s Punishment," Dynamic Games and Applications, Springer, vol. 12(2), pages 677-688, June.
    6. Lotito, Gianna & Migheli, Matteo & Ortona, Guido, 2019. "Some Experimental Evidence on Type Stability and Response Times," Department of Economics and Statistics Cognetti de Martiis. Working Papers 201919, University of Turin.

    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. Jean Rabanal & Daniel Friedman, 2014. "Incomplete Information, Dynamic Stability and the Evolution of Preferences: Two Examples," Dynamic Games and Applications, Springer, vol. 4(4), pages 448-467, December.
    2. Sandholm, William H., 2015. "Population Games and Deterministic Evolutionary Dynamics," Handbook of Game Theory with Economic Applications,, Elsevier.
    3. Jonathan Newton, 2018. "Evolutionary Game Theory: A Renaissance," Games, MDPI, vol. 9(2), pages 1-67, May.
    4. Ratul Lahkar & Vinay Ramani, 2021. "An Evolutionary Approach to Pollution Control in Competitive Markets," Working Papers 68, Ashoka University, Department of Economics.
    5. Sarvesh Bandhu & Ratul Lahkar, 2023. "Evolutionary robustness of dominant strategy implementation," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 76(2), pages 685-721, August.
    6. Dharini Hingu & K. S. Mallikarjuna Rao & A. J. Shaiju, 2018. "Evolutionary Stability of Polymorphic Population States in Continuous Games," Dynamic Games and Applications, Springer, vol. 8(1), pages 141-156, March.
    7. Ratul Lahkar & Vinay Ramani, 2022. "An Evolutionary Approach to Pollution Control in Competitive Markets," Dynamic Games and Applications, Springer, vol. 12(3), pages 872-896, September.
    8. Ratul Lahkar, 2020. "Convergence to Walrasian equilibrium with minimal information," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 15(3), pages 553-578, July.
    9. Dharini Hingu, 2020. "Asymptotic stability of strongly uninvadable sets," Annals of Operations Research, Springer, vol. 287(2), pages 737-749, April.
    10. Cheung, Man-Wah, 2014. "Pairwise comparison dynamics for games with continuous strategy space," Journal of Economic Theory, Elsevier, vol. 153(C), pages 344-375.
    11. Lahkar, Ratul & Mukherjee, Saptarshi, 2021. "Evolutionary implementation in aggregative games," Mathematical Social Sciences, Elsevier, vol. 109(C), pages 137-151.
    12. Lahkar, Ratul & Mukherjee, Saptarshi, 2019. "Evolutionary implementation in a public goods game," Journal of Economic Theory, Elsevier, vol. 181(C), pages 423-460.
    13. Cheung, Man-Wah, 2016. "Imitative dynamics for games with continuous strategy space," Games and Economic Behavior, Elsevier, vol. 99(C), pages 206-223.
    14. Sivan Frenkel & Yuval Heller & Roee Teper, 2018. "The Endowment Effect As Blessing," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 59(3), pages 1159-1186, August.
    15. Lahkar, Ratul, 2019. "Elimination of non-individualistic preferences in large population aggregative games," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 150-165.
    16. Lahkar, Ratul & Mukherjee, Sayan & Roy, Souvik, 2022. "Generalized perturbed best response dynamics with a continuum of strategies," Journal of Economic Theory, Elsevier, vol. 200(C).
    17. RatulLahkar & Sayan Mukherjee & Souvik Roy, 2021. "Generalized Perturbed Best Response Dynamics with a Continuum of Strategies," Working Papers 51, Ashoka University, Department of Economics.
    18. Dai Zusai, 2018. "Evolutionary dynamics in heterogeneous populations: a general framework for an arbitrary type distribution," Papers 1805.04897, arXiv.org, revised May 2019.
    19. Roee Teper, 2014. "The Endowment Effect as a Blessing," Working Paper 5862, Department of Economics, University of Pittsburgh.
    20. Ratul Lahkar & Sayan Mukherjee & Souvik Roy, 2022. "A Deterministic Approximation Approach to the Continuum Logit Dynamic with an Application to Supermodular Games," Working Papers 79, Ashoka University, Department of Economics.

    More about this item

    Keywords

    Population games; Evolutionary dynamics; Corporate social responsibility; Mixed oligopoly; Public goods;
    All these keywords.

    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
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets
    • L21 - Industrial Organization - - Firm Objectives, Organization, and Behavior - - - Business Objectives of the Firm
    • M14 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Administration - - - Corporate Culture; Diversity; Social Responsibility
    • H41 - Public Economics - - Publicly Provided Goods - - - Public Goods

    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:spr:dyngam:v:7:y:2017:i:1:d:10.1007_s13235-015-0164-0. 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.