## Abstract

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.

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## References

Nash, J., Non-cooperative Games,

*Ann. Math.*, 1951, no. 54, pp. 286–295.Cournot, A.A.,

*Researches into the Mathematical Principles of the Theory of Wealth*, London: Hafner, 1960.Bowley, A.L.,

*The Mathematical Groundwork of Economics*, Oxford: Oxford Univ. Press, 1924.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.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.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.Vasin, A., Game-Theoretic Study of Electricity Market Mechanisms,

*Procedia Comput. Sci.*, 2014, no. 31, pp. 124–132.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.Geraskin, M.I., Game-Theoretic Analysis of Stackelberg Oligopoly with Arbitrary Rank Reflexive Behavior of Agents,

*Kybern.*, 2017, no. 46 (6), pp. 1052–1067.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.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.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.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.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.Stackelberg, H.,

*Market Structure and Equilibrium*, Berlin: Springer-Verlag, 2011, 1st ed.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.Corchyn, L.C., Comparative Statics for Aggregative Games: The Strong Concavity Case,

*Math. Social Sci.*, 1994, vol. 28 (3), pp. 151–165.Possajennikov, A., Conjectural Variations in Aggregative Games: An Evolutionary Perspective,

*Math. Social Sci.*, 2015, no. 77, pp. 55–61.Walters, A.A., Production and Cost Functions: an Econometric Survey,

*Econometrica*, 1963, vol. 31, no. 1, pp. 23–44.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.Korn, G. and Korn, T.,

*Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review*, New York: McGraw-Hill, 1968.Petrov, I.B. and Lobanov, A.I.,

*Lektsii po vychislitel’noi matematike*(Lectures on Computational Mathematics), Moscow: BINOM, 2006.Varah, J.M., A Lower Bound for the Smallest Singular Value of a Matrix,

*Linear Algebra Appl.*, 1975, vol. 11(1), pp. 3–5.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.

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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.

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Geraskin, M.I. The Properties of Conjectural Variations in the Nonlinear Stackelberg Oligopoly Model.
*Autom Remote Control* **81**, 1051–1072 (2020). https://doi.org/10.1134/S0005117920060089

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DOI: https://doi.org/10.1134/S0005117920060089

### Keywords

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