Skip to main content
SpringerLink
Log in

To the Analytical Investigation of the Convergence Conditions of the Processes of Reflexive Collective Behavior in Oligopoly Models

  • CONTROL IN SOCIAL ECONOMIC SYSTEMS
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

A study of the dynamics of the collective behavior of interconnected rational agents under conditions of incomplete information is presented. Statements are proved that permit one to unify the conditions for convergence to equilibrium for two trajectories of the response of agents to the expected actions of the environment: (1) “classic” optimization behavior, in which agents try to choose optimal responses, ignoring possible current “negative states;” (2) behavior that guarantees each agent motion towards the objective and nonnegative current states. The application of the results obtained in the Cournot and Stackelberg oligopoly models with reflexive behavior of agents is shown. Sufficient conditions for the convergence of processes of collective behavior in an oligopoly with an arbitrary number of Stackelberg leaders are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Novikov, D.A. and Chkhartishvili, A.G., Reflexion and Control: Mathematical Models, Leiden: CRC Press, 2014.

    Book  Google Scholar 

  2. Novikov, D., Korepanov, V., and Chkhartishvili, A., Reflexion in mathematical models of decision-making, Int. J. Parallel Emerg. Distrib. Syst., 2018, vol. 33, no. 3, pp. 319–335.

    Article  Google Scholar 

  3. Cournot, A., Researches into the Mathematical Principles of the Theory of Wealth, London: Hafner, 1960 (Original 1838).

  4. Stackelberg, H., Market Structure and Equilibrium: 1st Ed. Translation into English, Basin–Urch&Hill: Springer, 2011 (Original 1934).

  5. The Handbook of Experimental Economics, Kagel, J. and Roth, A., Eds., Princeton: Princeton Univ. Press, 1995.

  6. Wright, J. and Leyton-Brown, K., Beyond equilibrium: predicting human behavior in normal form games, in Proc. Conf. Assoc. Adv. Artif. Intell. (AAAI-10), 2010, pp. 461–473.

  7. Aizenberg, N.I., Zorkal’tsev, V.I., and Mokryi, I.V., Study of unsteady oligopoly markets, J. Appl. Ind. Math., 2017, vol. 11, no. 1, pp. 8–16.

    Article  MathSciNet  Google Scholar 

  8. Geras’kin, M.I. and Chkhartishvili, A.G., Analysis of game-theoretic models of an oligopoly market under constrains on the capacity and competitiveness of agents, Autom. Remote Control, 2017, vol. 78, no. 11, pp. 2025–2038.

    Article  MathSciNet  Google Scholar 

  9. Opoitsev, V.I., Ravnovesie i ustoichivost’ v modelyakh kollektivnogo povedeniya (Equilibrium and Stability in Models of Collective Behavior), Moscow: Nauka, 1977.

    Google Scholar 

  10. Malishevskii, A.V., Kachestvennye modeli v teorii slozhnykh sistem (Qualitative Models in the Theory of Complex Systems), Moscow: Nauka, 1998.

    Google Scholar 

  11. Kalashnikov, V.V., Bulavsky, V.A., and Kalashnykova, N.I., Existence of the Nash-optimal strategies in the meta-game, Studies Syst., Decision Control, 2018, vol. 100, pp. 95–100.

    Article  Google Scholar 

  12. Algazin, G.I. and Algazina, D.G., Collective behavior in the Stackelberg model under incomplete information, Autom. Remote Control, 2017, vol. 78, no. 9, pp. 1619–1630.

    Article  MathSciNet  Google Scholar 

  13. Dusouchet, O.M., Static Cournot–Nash equilibria and strategic reflective games of oligopoly: a case of linear functions of demand and costs, Ekon. Zh. Vyssh. Shkola Ekon., 2006, no. 1, pp. 3–32.

  14. Algazin, G.I. and Algazina, Yu.G., Reflexive dynamics in the Cournot oligopoly under uncertainty, Autom. Remote Control, 2020, vol. 81, no. 2, pp. 345–359.

    Article  Google Scholar 

  15. Algazin, G.I. and Algazina, D.G., Reflexive processes and equilibrium in an oligopoly model with a leader, Autom. Remote Control, 2020, vol. 81, no. 7, pp. 1258–1270.

    Article  MathSciNet  Google Scholar 

  16. Samarskii, A.A. and Gulin, A.V., Chislennye metody (Numerical Methods), Moscow: Nauka, 1989.

    Google Scholar 

  17. Belitskii, G.R. and Lyubich, Yu.I., Normy matrits i ikh prilozheniya (Norms of Matrices and Applications), Kiev: Nauk. Dumka, 1984.

    Google Scholar 

  18. Ueda, M., Effect of information asymmetry in Cournot duopoly game with bounded rationality, Appl. Math. Comput., 2019, vol. 362, p. 124535.

    MathSciNet  MATH  Google Scholar 

  19. Elsadany, A.A., Dynamics of a Cournot duopoly game with bounded rationality based on relative profit maximization, Appl. Math. Comput., 2017, vol. 294, pp. 253–263.

    MathSciNet  MATH  Google Scholar 

  20. Askar, S. and Simos, T., Tripoly Stackelberg game model: one leader versus two followers, Appl. Math. Comput., 2018, vol. 328, pp. 301–311.

    MathSciNet  MATH  Google Scholar 

  21. Wu, R. and Van Gorder, R.A., Nonlinear dynamics of discrete time multi-level leader–follower games, Appl. Math. Comput., 2018, vol. 320, pp. 240–250.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Altai State University, Barnaul, 656049, Russia

    G. I. Algazin

  2. Polzunov Altai State Technical University, Barnaul, 656038, Russia

    Yu. G. Algazina

Authors
  1. G. I. Algazin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. G. Algazina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to G. I. Algazin or Yu. G. Algazina.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Algazin, G.I., Algazina, Y.G. To the Analytical Investigation of the Convergence Conditions of the Processes of Reflexive Collective Behavior in Oligopoly Models. Autom Remote Control 83, 367–388 (2022). https://doi.org/10.1134/S0005117922030067

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117922030067

Keywords

Search

Navigation