Skip to main content

The Properties of Conjectural Variations in the Nonlinear Stackelberg Oligopoly Model


The game-theoretic problem of choosing optimal strategies for oligopoly market agents with linear demand functions and nonlinear cost functions is considered. The conjectural variations of each agent, i.e., the expected responses (changes in actions) of his counteragents that optimize their utility functions, are studied. Formulas for calculating the conjectural variations of each agent and also the sum of the conjectural variations of all agents in the environment of each agent are derived. The signs of conjectural variations under an arbitrary level of Stackelberg leadership are analyzed. The following properties of conjectural variations are established: 1) the variations are negative if the cost functions of all environmental agents are either convex or concave; 2) the variations are positive if the agents with concave cost functions (the ones with the positive scale effect) prevail in the environment over the agents with convex cost functions (the ones with the negative scale effect). The sum of the agent’s conjectural variations is: 1) negative and its magnitude is bounded above by 1 if the environmental agents mainly have convex cost functions; 2) positive and unlimited if the agents with concave cost functions prevail in the environment.

This is a preview of subscription content, access via your institution.


  1. Nash, J., Non-cooperative Games, Ann. Math., 1951, no. 54, pp. 286–295.

  2. Cournot, A.A., Researches into the Mathematical Principles of the Theory of Wealth, London: Hafner, 1960.

    MATH  Google Scholar 

  3. Bowley, A.L., The Mathematical Groundwork of Economics, Oxford: Oxford Univ. Press, 1924.

    MATH  Google Scholar 

  4. Karmarkar, U.S. and Rajaram, K., Aggregate Production Planning for Process Industries under Oligopolistic Competition, Eur. J. Oper. Res., 2012, no. 223 (3), pp. 680–689.

    MathSciNet  Article  Google Scholar 

  5. Ledvina, A. and Sigar, R., Oligopoly Games under Asymmetric Costs and an Application to Energy Production, Math. Finan. Econ., 2012, no. 6 (4), pp. 261–293.

    MathSciNet  Article  Google Scholar 

  6. Currarini, S. and Marini, M.A., Sequential Play and Cartel Stability in Cournot Oligopoly, Appl. Math. Sci., 2013, no. 7 (1–4), pp. 197–200.

    MathSciNet  Google Scholar 

  7. Vasin, A., Game-Theoretic Study of Electricity Market Mechanisms, Procedia Comput. Sci., 2014, no. 31, pp. 124–132.

  8. Sun, F., Liu, B., Hou, F., Gui, L., and Chen, J., Cournot Equilibrium in the Mobile Virtual Network Operator Oriented Oligopoly Offloading Market, Proc. 2016 IEEE Int. Conf. Commun. (ICC 2016), Kuala Lumpur, Malaysia, 2016, no. 7511340.

  9. Geraskin, M.I., Game-Theoretic Analysis of Stackelberg Oligopoly with Arbitrary Rank Reflexive Behavior of Agents, Kybern., 2017, no. 46 (6), pp. 1052–1067.

    Article  Google Scholar 

  10. Geraskin, M., Equilibria in the Stackelberg Oligopoly Reflexive Games with Different Marginal Costs of Agents, Int. Game Theory Rev., 2019, vol. 21, no. 4, pp. 1–22.

    MathSciNet  Article  Google Scholar 

  11. Naimzada, A.K. and Sbragia, L., Oligopoly Games with Nonlinear Demand and Cost Functions: Two Boundedly Rational Adjustment Processes, Chaos, Solit. Fractal., 2006, no. 29 (3), pp. 707–722.

    MathSciNet  Article  Google Scholar 

  12. Askar, S. and Alnowibet, K., Nonlinear Oligopolistic Game with Isoelastic Demand Function: Rationality and Local Monopolistic Approximation, Chaos, Solit. Fractal., 2016, no. 84, pp. 15–22.

  13. Naimzada, A. and Tramontana, F., Two Different Routes to Complex Dynamics in a Heterogeneous Triopoly Game, J. Differ. Eq. Appl., 2015, no. 21 (7), pp. 553–563.

    MathSciNet  Article  Google Scholar 

  14. Cavalli, F., Naimzada, A., and Tramontana, F., Nonlinear Dynamics and Global Analysis of a Geterogeneous Cournot Duopoly with a Local Monopolistic Approach Versus a Gradient Rule with Endogenous Reactivity, Commun. Nonlin. Sci. Numer. Simulat., 2015, no. 23 (1–3), pp. 245–262.

    Article  Google Scholar 

  15. Stackelberg, H., Market Structure and Equilibrium, Berlin: Springer-Verlag, 2011, 1st ed.

    Book  Google Scholar 

  16. Geraskin, M.I. and Chkhartishvili, A.G., Game-Theoretic Models of an Oligopoly Market with Nonlinear Agent Cost Functions, Autom. Remote Control, 2017, vol. 78, no. 9, pp. 1631–1650.

    MathSciNet  Article  Google Scholar 

  17. Corchyn, L.C., Comparative Statics for Aggregative Games: The Strong Concavity Case, Math. Social Sci., 1994, vol. 28 (3), pp. 151–165.

    MathSciNet  Article  Google Scholar 

  18. Possajennikov, A., Conjectural Variations in Aggregative Games: An Evolutionary Perspective, Math. Social Sci., 2015, no. 77, pp. 55–61.

  19. Walters, A.A., Production and Cost Functions: an Econometric Survey, Econometrica, 1963, vol. 31, no. 1, pp. 23–44.

    MATH  Google Scholar 

  20. Geraskin, M.I., Modeling Reflexion in the Non-Linear Model of the Stakelberg Three-Agent Oligopoly for the Russian Telecommunication Market, Autom. Remote Control, 2018, vol. 79, no. 5, pp. 841–859.

    MathSciNet  Article  Google Scholar 

  21. Korn, G. and Korn, T., Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, New York: McGraw-Hill, 1968.

    MATH  Google Scholar 

  22. Petrov, I.B. and Lobanov, A.I., Lektsii po vychislitel’noi matematike (Lectures on Computational Mathematics), Moscow: BINOM, 2006.

    Google Scholar 

  23. Varah, J.M., A Lower Bound for the Smallest Singular Value of a Matrix, Linear Algebra Appl., 1975, vol. 11(1), pp. 3–5.

    MathSciNet  Article  Google Scholar 

  24. Reddy Rachapalli, S. and Kulshreshtha, P., Evolutionarily Stable Conjectures and Social Optimality in Oligopolies, Theoret. Econ. Lett., 2013, vol. 3, no. 1, pp. 12–18.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to M. I. Geraskin.

Additional information

This paper was recommended for publication by M. V. Gubko, a member of the Editorial Board

Russian Text © The Author(s), 2020, published in Avtomatika i Telemekhanika, 2020, No. 6, pp. 105–130.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Geraskin, M.I. The Properties of Conjectural Variations in the Nonlinear Stackelberg Oligopoly Model. Autom Remote Control 81, 1051–1072 (2020).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • oligopoly
  • Stackelberg game
  • power cost function
  • multilevel leadership