Convergence in Norm of Collective Behavior Dynamics in the Reflexive Model of Oligopoly with Leaders

Cover Page

Cite item

Full Text

Abstract

A model of oligopoly with an arbitrary number of rational agents that are reflexive according to Cournot or Stackelberg, under the conditions of incomplete information for the classical case of linear functions of costs and demand is considered. The problem of achieving equilibrium based on mathematical modeling agents' decision-making processes is investigated. Works in this direction are relevant due to the importance of understanding the processes in real markets and the convergence of theoretical models with them. In the framework of a dynamic model of reflexive collective behavior, each agent at each moment adjusts its output, making a step in the direction of output maximizing its profit under the expected choice of competitors. The permissible step value is set by the range. This article sets and solves the problem of finding the ranges of permissible steps of agents, which are formulated as conditions that guarantee the convergence of dynamics to equilibrium. The novelty of the study is determined by the use of the norm of the error transition matrix from the t-th to (t+1)-moment of time as a criterion of the dynamics convergence. It is shown that the dynamics converge if the norm is less than unity, starting at some point in time, and the failure to fulfill this criterion especially manifests itself in multidirectional choice, when some agents choose "big" steps towards their current goals, while others, on the contrary, choose "small" steps. Failure to meet the criterion also increases as the market grows. The general conditions for the ranges of convergence of dynamics for an arbitrary number of agents are established, and a method for constructing the maximum such ranges is proposed, which also constitutes the novelty of the study. The results of solving the above problems for particular cases of oligopoly, which are the most widespread in practice, are presented.

About the authors

G. I Algazin

Altai State University

Email: algaz46@yandex.ru
Lenin Av. 61

D. G Algazina

Altai State University

Email: darya.algazina@mail.ru
Lenin Av. 61

References

  1. Cournot A. Researches into the Mathematical Principles of the Theory of Wealth // London: Hafner. 1960. 235 p.
  2. Stackelberg H. Market Structure and Equilibrium / Transl. into English by Basin D., Urch L. & Hill. R. // New York, Springer. 2011. 134 p.
  3. Zewde A.B., Kassa S.M. Multilevel Multi-Leader Multiple-Follower Games with Nonseparable Objectives and Shared Constraints // Computational Management Science. 2021. vol. 18(4). pp. 455–475.
  4. Wu R., Van Gorder R.A. Nonlinear Dynamics of Discrete Time Multi-Level Leader-Follower Games // Applied Mathematics and Computation. 2018. vol. 320. pp. 240–250.
  5. Geras’kin M.I. The Properties of Conjectural Variations in the Nonlinear Stackelberg Oligopoly Model // Automation and Remote Control. 2020. vol. 81. no. 6. pp. 1051–1072.
  6. Geras’kin M.I. Approximate Calculation of Equilibria in the Nonlinear Stackelberg Oligopoly Model: A Linearization Based Approach // Automation and Remote Control. 2020. vol. 81 no. 9. pp. 1659–1678.
  7. Castiglioni M., Marchesi A., Gatti N. Committing to Correlated Strategies with Multiple Leaders // Artificial Intelligence. 2021. vol. 300. doi: 10.1016/j.artint.2021.103549.
  8. Zewde A.B., Kassa S.M. Hierarchical Multilevel Optimization with Multiple-Leaders Multiple-Followers Setting and Nonseparable Objectives // RAIRO – Operations Research. 2021. vol. 55(5). pp. 2915–2939.
  9. Алгазин Г.И., Алгазина Д.Г. Моделирование динамики коллективного поведения в рефлексивной игре с произвольным числом лидеров // Информатика и автоматизация. 2022. Т. 21. № 2. С. 339–375.
  10. Alcantara-Jiménez G., Clempner J.B. Repeated Stackelberg Security Games: Learning with Incomplete State Information // Reliability Engineering and System Safety. 2020. vol. 195. doi: 10.1016/j.ress.2019.106695.
  11. Algazin G.I., Algazina D.G. Reflexive Processes and Equilibrium in an Oligopoly Model with a Leader // Automation and Remote Control. 2020. vol. 81. no. 7. pp. 1258–1270.
  12. Скаржинская Е.М., Цуриков В.И. Влияние личностных качеств агентов на экзогенное формирование лидерства по Штакельбергу в модели коллективных действий // Экономика и математические методы. 2022. № 4. С. 113–122.
  13. Novikov D.A., Chkhartishvili A.G. Reflexion and Control: Mathematical Models // Leiden: CRC Press. 2014. 298 p.
  14. Nash J. Non-Cooperative Games // Annals of Mathematics. 1951. no. 54. pp. 286–295.
  15. The Handbook of Experimental Economics / Ed. by Kagel J. and Roth A. // Princeton: Princeton University Press. 1995. 744 p.
  16. Wright J., Leyton-Brown K. Beyond Equilibrium: Predicting Human Behavior in Normal Form Games // Proceedings of Conference on Associations for the Advancement of Artificial Intelligence (AAAI-10). 2010. pp. 461–473.
  17. Askar S., Simos T. Tripoly Stackelberg Game Model: One Leader Versus Two Followers // Applied Mathematics and Computation. 2018. vol. 328. pp. 301–311.
  18. Askar S. On Complex Dynamics of Cournot-Bertrand Game with Asymmetric Market Information // Applied Mathematics and Computation. 2021. vol. 393(3). doi: 10.1016/j.amc.2020.125823.
  19. Korgin N.A., Korepanov V.O. Nash Bargaining Solution as Negotiation Concept for Resource Allocation Problem: Analysis of Experimental Data // Contributions to Game Theory and Management. 2020. no. 13. pp. 207–217.
  20. Fedyanin D.Н. An Example of Reflexive Analysis of a Game in Normal Form / Ed. by Petrosyan L., Mazalov V., Zenkevich N. // Frontiers of Dynamic Games. Static & Dynamic Game Theory: Foundations & Applications. Birkhäuser, Cham. 2019. pp. 1–11. doi: 10.1007/978-3-030-23699-1_1.
  21. Опойцев В.И. Равновесие и устойчивость в моделях коллективного поведения // М.: Наука. 1977. 248 c.
  22. Новиков Д.А. Модели динамики психических и поведенческих компонент деятельности в коллективном принятии решений // Управление большими системами: М. ИПУ РАН. 2020. № 85. С. 206–237.
  23. Корепанов В.О. Модели рефлексивного группового поведения и управления // М.: ИПУ РАН. 2011. 127 с.
  24. Algazin G.I., Algazina Yu.G. Reflexive Dynamics in the Cournot Oligopoly under Uncertainty // Automation and Remote Control. 2020. vol. 81. no. 2. pp. 345–359.
  25. Geraskin M.I. Reflexive Analysis of Equilibria in a Triopoly Game with Linear Cost Functions of the Agents’ // Automation and Remote Control. 2022. vol. 83. no. 3. pp. 389–406.
  26. Askar S.S., Elettrebybc M.F. The Impact of Cost Uncertainty on Cournot Oligopoly Games // Applied Mathematics and Computation. 2017. vol. 312. pp. 169–176.
  27. Ueda M. Effect of Information Asymmetry in Cournot Duopoly Game with Bounded Rationality // Applied Mathematics and Computation. 2019. vol. 362. doi: 10.1016/j.amc.2019.06.049.124535.
  28. Long J., Huang H. A Dynamic Stackelberg-Cournot Duopoly Model with Heterogeneous Strategies through One-Way Spillovers // Discrete Dynamics in Nature and Society. 2020. vol. 2. pp. 1–11.
  29. Elsadany A.A. Dynamics of a Cournot Duopoly Game with Bounded Rationality Based on Relative Profit Maximization // Applied Mathematics and Computation. 2017. vol. 294. pp. 253–263.
  30. Al-Khedhairi A. Dynamical Study of Competition Cournot-like Duopoly Games Incorporating Fractional Order Derivatives and Seasonal Influences // International Journal of Nonlinear Sciences and Numerical Simulation. 2020. vol. 21. pp. 339–359.
  31. Algazin G.I., Algazina Yu.G. To the Analytical Investigation of the Convergence Conditions of the Processes of Reflexive Collective Behavior in Oligopoly Models // Automation and Remote Control. 2022. vol. 83. no. 3. pp. 367–388.
  32. Самарский А.А., Гулин А.В. Численные методы // М.: Наука. 1989. 432 с.
  33. Белицкий Г.Р., Любич Ю.И. Нормы матриц и их приложения // Киев: Наукова думка. 1984. 158 с.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).