IDEAS home Printed from https://ideas.repec.org/a/wsi/igtrxx/v20y2018i02ns0219198917500335.html
   My bibliography  Save this article

On Lyapunov Game Theory Equilibrium: Static and Dynamic Approaches

Author

Listed:
  • Julio B. Clempner

    (Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, School of Physics and Mathematics, National Polytechnic Institute, Luis Enrique Erro S/N, San Pedro Zacatenco, Del. Gustavo A. Madero 07738, Mexico City, Mexico)

Abstract

This paper suggests a game theory problem in which any feasible solution is based on the Lyapunov theory. The problem is analyzed in the static and dynamic cases. Some properties of Nash equilibria such as existence and stability are derived naturally from the Lyapunov theory. Remarkable is that every asymptotically stable equilibrium point (Nash equilibrium point) admits a Lyapunov-like function and if a Lyapunov-like function exists it converges to a Nash/Lyapunov equilibrium point. We define a Lyapunov-like function as an Lp-norm from the multiplayer objective function to the utopia minimum as a cost function. We propose multiple metrics to find the Nash/Lyapunov equilibrium and the strong Nash/Lyapunov equilibrium. Finding a Nash/Lyapunov equilibrium is reduced to the minimization problem of the Lyapunov-like function. We prove that the equilibrium point properties of Nash and Lyapunov meet in game theory. In order to validate the contributions of the paper, we present a numerical example.

Suggested Citation

  • Julio B. Clempner, 2018. "On Lyapunov Game Theory Equilibrium: Static and Dynamic Approaches," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(02), pages 1-14, June.
    • Handle: RePEc:wsi:igtrxx:v:20:y:2018:i:02:n:s0219198917500335
    DOI: 10.1142/S0219198917500335
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219198917500335
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219198917500335?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. Margaret E. Slade, 1989. "The Fictitious Payoff Function: Two Applications to Dynamic Games," Annals of Economics and Statistics, GENES, issue 15-16, pages 193-216.
    2. Ricardo Azevedo Araujo & Helmar Nunes Moreira, 2014. "Lyapunov stability in an evolutionary game theory model of the labour market," Economia, ANPEC - Associação Nacional dos Centros de Pós-Graduação em Economia [Brazilian Association of Graduate Programs in Economics], vol. 15(1), pages 41-53.
    3. Monderer, Dov & Shapley, Lloyd S., 1996. "Fictitious Play Property for Games with Identical Interests," Journal of Economic Theory, Elsevier, vol. 68(1), pages 258-265, January.
    4. repec:adr:anecst:y:1989:i:15-16:p:09 is not listed on IDEAS
    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. Julio B. Clempner & Alexander S. Poznyak, 2021. "Analytical Method for Mechanism Design in Partially Observable Markov Games," Mathematics, MDPI, vol. 9(4), pages 1-15, February.
    2. Ruda Zhang & Roger Ghanem, 2020. "Multi-market Oligopoly of Equal Capacity," Papers 2012.06742, arXiv.org.
    3. Kristal K. Trejo & Ruben Juarez & Julio B. Clempner & Alexander S. Poznyak, 2023. "Non-Cooperative Bargaining with Unsophisticated Agents," Computational Economics, Springer;Society for Computational Economics, vol. 61(3), pages 937-974, 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. Anthonisen, Niels, 1997. "On the Convergence of Beliefs within Populations in Games with Learning," Journal of Economic Theory, Elsevier, vol. 76(1), pages 169-184, September.
    2. 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.
    3. Rossella Argenziano & Itzhak Gilboa, 2012. "History as a coordination device," Theory and Decision, Springer, vol. 73(4), pages 501-512, October.
    4. Julide Yazar, 2024. "Bayesian Fictitious Play in Oligopoly: The Case of Risk-Averse Agents," Games, MDPI, vol. 15(6), pages 1-15, November.
    5. Hopkins, Ed, 1999. "Learning, Matching, and Aggregation," Games and Economic Behavior, Elsevier, vol. 26(1), pages 79-110, January.
    6. Sobel, Joel, 2000. "Economists' Models of Learning," Journal of Economic Theory, Elsevier, vol. 94(2), pages 241-261, October.
    7. Jiequn Han & Ruimeng Hu & Jihao Long, 2020. "Convergence of Deep Fictitious Play for Stochastic Differential Games," Papers 2008.05519, arXiv.org, revised Mar 2021.
    8. Hu, Yu & Chi, Yuanying & Zhou, Wenbing & Li, Jialin & Wang, Zhengzao & Yuan, Yongke, 2023. "The interactions between renewable portfolio standards and carbon emission trading in China: An evolutionary game theory perspective," Energy, Elsevier, vol. 271(C).
    9. Monderer, Dov & Sela, Aner, 1997. "Fictitious play and- no-cycling conditions," Sonderforschungsbereich 504 Publications 97-12, Sonderforschungsbereich 504, Universität Mannheim;Sonderforschungsbereich 504, University of Mannheim.
    10. Stefanos Leonardos & Costis Melolidakis, 2018. "On the Commitment Value and Commitment Optimal Strategies in Bimatrix Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-28, September.
    11. In, Younghwan, 2014. "Fictitious play property of the Nash demand game," Economics Letters, Elsevier, vol. 122(3), pages 408-412.
    12. Narges TALEBIMOTLAGH & Farzad HASHEMZADEH & Amir RIKHTEHGAR GHIASI & Sehraneh GHAEMI, 2017. "A Novel Method of Modeling Dynamic Evolutionary Game with Rational Agents for Market Forecasting," ECONOMIC COMPUTATION AND ECONOMIC CYBERNETICS STUDIES AND RESEARCH, Faculty of Economic Cybernetics, Statistics and Informatics, vol. 51(1), pages 281-302.
    13. Hon-Snir, Shlomit & Monderer, Dov & Sela, Aner, 1998. "A Learning Approach to Auctions," Journal of Economic Theory, Elsevier, vol. 82(1), pages 65-88, September.
    14. Joseph Abdou & Nikolaos Pnevmatikos & Marco Scarsini, 2014. "Uniformity and games decomposition," Documents de travail du Centre d'Economie de la Sorbonne 14084r, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Mar 2017.
    15. Ding, Zhanwen & Wang, Qiao & Cai, Chaoying & Jiang, Shumin, 2014. "Fictitious play with incomplete learning," Mathematical Social Sciences, Elsevier, vol. 67(C), pages 1-8.
    16. Jacques Durieu & Philippe Solal, 2012. "Models of Adaptive Learning in Game Theory," Chapters, in: Richard Arena & Agnès Festré & Nathalie Lazaric (ed.), Handbook of Knowledge and Economics, chapter 11, Edward Elgar Publishing.
    17. Ruben Juarez & Rajnish Kumar, 2013. "Implementing efficient graphs in connection networks," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(2), pages 359-403, October.
    18. Berger, Ulrich, 2005. "Fictitious play in 2 x n games," Journal of Economic Theory, Elsevier, vol. 120(2), pages 139-154, February.
    19. Morris, Stephen & Ui, Takashi, 2004. "Best response equivalence," Games and Economic Behavior, Elsevier, vol. 49(2), pages 260-287, November.
    20. Duffy, John & Hopkins, Ed, 2005. "Learning, information, and sorting in market entry games: theory and evidence," Games and Economic Behavior, Elsevier, vol. 51(1), pages 31-62, April.

    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:wsi:igtrxx:v:20:y:2018:i:02:n:s0219198917500335. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/igtr/igtr.shtml .

    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.