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On Lyapunov Game Theory Equilibrium: Static and Dynamic Approaches

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  • 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
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    References listed on IDEAS

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    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. repec:adr:anecst:y:1989:i:15-16:p:09 is not listed on IDEAS
    4. 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.
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    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.

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