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Stability in games with continua of equilibria

Author

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  • Sebastian Bervoets

    (AMSE - Aix-Marseille Sciences Economiques - EHESS - École des hautes études en sciences sociales - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique)

  • Mathieu Faure

    (AMSE - Aix-Marseille Sciences Economiques - EHESS - École des hautes études en sciences sociales - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique)

Abstract

The stability of Nash equilibria has often been studied by examining the asymptotic behavior of the best-response dynamics. This is generally done in games where interactions are global and equilibria are isolated. In this paper, we analyze stability in contexts where interactions are local and where there are continua of equilibria. We focus on the public good game played on a network, where the set of equilibria is known to depend on the network structure (Bramoullé and Kranton, 2007), and where, as we show, continua of equilibria often appear. We provide necessary and sufficient conditions for a component of Nash equilibria to be asymptotically stable vis-à-vis the best-response dynamics. Interestingly, we demonstrate that these conditions relate to the structure of the network in a simple way. We also provide corresponding results for several dynamical systems related to the best response.

Suggested Citation

  • Sebastian Bervoets & Mathieu Faure, 2019. "Stability in games with continua of equilibria," Post-Print hal-02021221, HAL.
    • Handle: RePEc:hal:journl:hal-02021221
    DOI: 10.1016/j.jet.2018.10.011
    Note: View the original document on HAL open archive server: https://amu.hal.science/hal-02021221
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    1. Kukushkin, Nikolai S., 2015. "Cournot tatonnement and potentials," Journal of Mathematical Economics, Elsevier, vol. 59(C), pages 117-127.
    2. Voorneveld, Mark, 2000. "Best-response potential games," Economics Letters, Elsevier, vol. 66(3), pages 289-295, March.
    3. Matthew O. Jackson & Brian W. Rogers & Yves Zenou, 2017. "The Economic Consequences of Social-Network Structure," Journal of Economic Literature, American Economic Association, vol. 55(1), pages 49-95, March.
    4. Yann Bramoull? & Rachel Kranton & Martin D'Amours, 2014. "Strategic Interaction and Networks," American Economic Review, American Economic Association, vol. 104(3), pages 898-930, March.
    5. R. D. Theocharis, 1960. "On the Stability of the Cournot Solution on the Oligopoly Problem," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 27(2), pages 133-134.
    6. Michel Benaim & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions II: Applications," Levine's Bibliography 784828000000000098, UCLA Department of Economics.
    7. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions; Part II: Applications," Working Papers hal-00242974, HAL.
    8. Franklin M. Fisher, 1961. "The Stability of the Cournot Oligopoly Solution: The Effects of Speeds of Adjustment and Increasing Marginal Costs," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 28(2), pages 125-135.
    9. al-Nowaihi, A. & Levine, P. L., 1985. "The stability of the cournot oligopoly model: A reassessment," Journal of Economic Theory, Elsevier, vol. 35(2), pages 307-321, August.
    10. Seade, Jesus, 1980. "The stability of cournot revisited," Journal of Economic Theory, Elsevier, vol. 23(1), pages 15-27, August.
    11. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
    12. F. H. Hahn, 1962. "The Stability of the Cournot Oligopoly Solution," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 29(4), pages 329-331.
    13. Bramoulle, Yann & Kranton, Rachel, 2007. "Public goods in networks," Journal of Economic Theory, Elsevier, vol. 135(1), pages 478-494, July.
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    Cited by:

    1. Bervoets, Sebastian & Faure, Mathieu, 2020. "Convergence in games with continua of equilibria," Journal of Mathematical Economics, Elsevier, vol. 90(C), pages 25-30.
    2. Nizar Allouch, 2017. "Aggregation in Networks," Studies in Economics 1718, School of Economics, University of Kent.
    3. Bayer, Péter & Herings, P. Jean-Jacques & Peeters, Ronald, 2021. "Farsighted manipulation and exploitation in networks," Journal of Economic Theory, Elsevier, vol. 196(C).
    4. Péter Bayer & György Kozics & Nóra Gabriella Szőke, 2020. "Best-Response Dynamics in Directed Network Games," CEU Working Papers 2020_1, Department of Economics, Central European University.
    5. Bayer, Péter & Herings, P. Jean-Jacques & Peeters, Ronald & Thuijsman, Frank, 2019. "Adaptive learning in weighted network games," Journal of Economic Dynamics and Control, Elsevier, vol. 105(C), pages 250-264.
    6. Papadimitriou, Christos & Peng, Binghui, 2023. "Public goods games in directed networks," Games and Economic Behavior, Elsevier, vol. 139(C), pages 161-179.
    7. Battigalli, Pierpaolo & Panebianco, Fabrizio & Pin, Paolo, 2023. "Learning and selfconfirming equilibria in network games," Journal of Economic Theory, Elsevier, vol. 212(C).
    8. Zenou, Yves & Bochet, Olivier & Faure, Mathieu & Long, Yan, 2020. "Perceived Competition in Networks," CEPR Discussion Papers 15582, C.E.P.R. Discussion Papers.
    9. Péter Bayer & György Kozics & Nóra Szőke, 2019. "Best-Response Dynamics in Directed Network Games," CEU Working Papers 2019_3, Department of Economics, Central European University.
    10. P'eter Bayer & Gyorgy Kozics & N'ora Gabriella SzH{o}ke, 2021. "Best-response dynamics in directed network games," Papers 2101.03863, arXiv.org.
    11. Bochet, Olivier & Faure, Mathieu & Long, Yan & Zenou, Yves, 2020. "Perceived Competition in Networks," CEPR Discussion Papers 15582, C.E.P.R. Discussion Papers.

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    More about this item

    Keywords

    Best-response dynamics; Public good games; Stability;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness
    • H41 - Public Economics - - Publicly Provided Goods - - - Public Goods

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