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Analyzing An Optimistic Attitude For The Leader Firm In Duopoly Models: A Strong Stackelberg Equilibrium Based On A Lyapunov Game Theory Approach

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Listed:
  • Julio B. CLEMPNER

    (Instituto Politécnico Nacional, Mexico)

  • Alexander S. POZNYAK

    (Center for Research and Advanced Studies)

Abstract

This paper presents a novel game theory approach for representing the Stackelberg dynamic duopoly models. The problem is fitted into a class of ergodic controllable finite Markov chains game. It is consider the case where a strong Stackelberg equilibrium point is convenient for both firms. We first analyze the best-reply dynamics of the Stackelberg duopoly model such that each firm’s best-reply strategies set agree with the set of maximizers of the opponents’ strategies. Then, the classical Stackelberg duopoly game is transformed into a potential game in terms of the Lyapunov theory. As a result, a duopoly model has the benefit that the best-reply dynamics results in a natural implementation of the behavior of a Lyapunov-like function. In addition, the strong equilibrium point properties of Stackelberg and Lyapunov meet in potential games. We validate the proposed method theoretically by computing the complexity and by a numerical experiment related to the duopoly model.

Suggested Citation

  • Julio B. CLEMPNER & Alexander S. POZNYAK, 2016. "Analyzing An Optimistic Attitude For The Leader Firm In Duopoly Models: A Strong Stackelberg Equilibrium Based On A Lyapunov Game Theory Approach," ECONOMIC COMPUTATION AND ECONOMIC CYBERNETICS STUDIES AND RESEARCH, Faculty of Economic Cybernetics, Statistics and Informatics, vol. 50(4), pages 41-60.
    • Handle: RePEc:cys:ecocyb:v:50:y:2016:i:4:p:41-60
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    References listed on IDEAS

    as
    1. Toshihiro Matsumura, 2003. "Stackelberg Mixed Duopoly with a Foreign Competitor," Bulletin of Economic Research, Wiley Blackwell, vol. 55(3), pages 275-287, July.
    2. Voorneveld, Mark, 2000. "Best-response potential games," Economics Letters, Elsevier, vol. 66(3), pages 289-295, March.
    3. Julio B. Clempner, 2015. "Setting Cournot Versus Lyapunov Games Stability Conditions and Equilibrium Point Properties," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 17(04), pages 1-10.
    4. Richard G. Harris & Elmer G. Wiens, 1980. "Government Enterprise: An Instrument for the Internal Regulation of Industry," Canadian Journal of Economics, Canadian Economics Association, vol. 13(1), pages 125-132, February.
    5. Breitmoser, Yves, 2012. "On the endogeneity of Cournot, Bertrand, and Stackelberg competition in oligopolies," International Journal of Industrial Organization, Elsevier, vol. 30(1), pages 16-29.
    6. de Fraja, Giovanni & Delbono, Flavio, 1990. "Game Theoretic Models of Mixed Oligopoly," Journal of Economic Surveys, Wiley Blackwell, vol. 4(1), pages 1-17.
    7. D. Dragone & L. Lambertini & A. Palestini, 2008. "A Class of Best-Response Potential Games," Working Papers 635, Dipartimento Scienze Economiche, Universita' di Bologna.
    8. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
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    Citations

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    Cited by:

    1. Herty, Michael & Steffensen, Sonja & Thünen, Anna, 2022. "Multiscale control of Stackelberg games," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 468-488.
    2. Julio B. Clempner, 2021. "A Proximal/Gradient Approach for Computing the Nash Equilibrium in Controllable Markov Games," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 847-862, March.
    3. Alcantara-Jiménez, Guillermo & Clempner, Julio B., 2020. "Repeated Stackelberg security games: Learning with incomplete state information," Reliability Engineering and System Safety, Elsevier, vol. 195(C).
    4. Julio B. Clempner, 2018. "Strategic Manipulation Approach for Solving Negotiated Transfer Pricing Problem," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 304-316, July.

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    More about this item

    Keywords

    Dynamic duopoly model; complexity; strong Stackelberg equilibrium; Lyapunov equilibrium; Stackelberg games; Lyapunov games; Markov decision process;
    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
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • E37 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Forecasting and Simulation: Models and Applications
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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