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

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  • 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 (Unpublished manuscript, 2001) public goods game and the Kopel et al.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
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    References listed on IDEAS

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    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. Jorgen W. Weibull, 1997. "Evolutionary Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262731215, April.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Cheung, Man-Wah, 2014. "Pairwise comparison dynamics for games with continuous strategy space," Journal of Economic Theory, Elsevier, vol. 153(C), pages 344-375.
    13. Kopel, Michael & Brand, Björn, 2012. "Socially responsible firms and endogenous choice of strategic incentives," Economic Modelling, Elsevier, vol. 29(3), pages 982-989.
    14. 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.
    15. 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.
    16. repec:hhs:iuiwop:487 is not listed on IDEAS
    17. 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.
    18. 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.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    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.

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    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

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