Skip to main content
SpringerLink
Log in

Adaptive rules for discrete-time Cournot games of high competition level markets

  • Original paper
  • Published:
Operational Research Aims and scope Submit manuscript

Abstract

For the Cournot-like oligopoly games of n firms, where the Cournot adjustment fails to converge, we propose adjustment processes originating from the family of the Moving Averages. In markets of linear demand, where firms have private and linear on quantities cost functions, these adaptive rules turn the games into discrete-time linear systems with delays. With an out of the box proof, we determine the least number of delays (m) that ensures the game of n players converges to its equilibrium. The Simple Moving Average rule (fixed number of delays) and the Cumulative Moving Average rule (constantly increasing number of delays), which is also known as “fictitious play”, are the main rules considered. Along with a hybrid rule, result of their combination, they are all studied for their convergent properties and compared in a bench-marking framework to indicate the different trajectories and identify their suitability in applications.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. Alternations of the adjustment process (continuous or discrete form), which are based on the speed of adaptation concept (Fisher 1961), are able to result convergence (Okuguchi 1970; Szidarovszky and Okuguchi 1988).

  2. System’s stability refers to global, uniform, asymptotic stability of the equilibrium point \({\mathbf {x}}^{*}\), i.e. for any \(\varepsilon >0\) there exists a \(\delta (\varepsilon )>0\) such that \(\| {\mathbf {x}}_{t_{0}}-{\mathbf {x}}^{*}\| <\delta\) implies \(\left\| {\mathbf {x}}_{t}-{\mathbf {x}}^{*}\right\| <\varepsilon ,\,\forall t\ge t_{0}\) and \(\lim\nolimits_{{t \to \infty }} {\mathbf{x}}_{t} = {\mathbf{x}}^{*}\), independently of \(t_{0}\) and \({\mathbf {x}}_{t_{0}}\) (Khalil 2002).

  3. Since matrix \({\mathbf {A}}\) is a scaled instance of \(\mathbf {A_G}\), one of its eigenvalues is dominant and all the other are strictly smaller than that, in absolute value (Sect. 2). With marginal stability, we mean the case where the system is neither asymptotically stable nor unstable, i.e. the case where the greatest magnitude of any of the eigenvalues is one and the multiplicity of this critical eigenvalue is one.

  4. Firms’ output can take non-negative values and this has been imposed in simulations; The nonlinearities that may occur (Cánovas et al. 2008) in the case of bounded action sets, given the existence of a Nash Equilibrium are out of the scope of this paper.

References

  • Ahmed E, Agiza H (1998) Dynamics of a cournot game with n-competitors. Chaos Solitons Fractals 9(9):1513–1517

    Article  Google Scholar 

  • Askar SS (2018) Duopolistic stackelberg game: investigation of complex dynamics and chaos control. Oper Res. https://doi.org/10.1007/s12351-018-0379-0

    Article  Google Scholar 

  • al Nowaihi A, Levine P (1985) The stability of the cournot oligopoly model: a reassessment. J Econ Theory 35(2):307–321

    Article  Google Scholar 

  • Bernhard P, Deschamps M (2017) On dynamic games with randomly arriving players. Dyn Games Appl 7(3):360–385

    Article  Google Scholar 

  • Cánovas JS, Puu T, Ruíz M (2008) The cournot-theocharis problem reconsidered. Chaos Solitons Fractals 37(4):1025–1039

    Article  Google Scholar 

  • Chrysanthopoulos N, Papavassilopoulos GP (2016) Learning optimal strategies in a stochastic game with partial information applied to electricity markets. In: Mediterranean conference on power generation, transmission, distribution and energy conversion (MedPower 2016), IET, pp 1–6

  • Chrysanthopoulos N, Papavassilopoulos GP (2016) A step closer to competition: a different concept of cournot duopoly under incomplete information. In: 10th international conference on game theory and management, SPbSU-ISDG, St. Petersburg, Russia

  • Daughety AF (2005) Cournot oligopoly: characterization and applications. Cambridge University Press, Cambridge

    Google Scholar 

  • Deschamps R (1975) An algorithm of game theory applied to the duopoly problem. Eur Econ Rev 6(2):187–194. https://doi.org/10.1016/0014-2921(75)90030-6

    Article  Google Scholar 

  • Doukas H (2013) Linguistic multicriteria decision making for energy systems: building the “re2s“ framework. Wiley Interdiscip Rev Energy Environ 2(5):571–585. https://doi.org/10.1002/wene.65

    Article  Google Scholar 

  • Doukas H, Patlitzianas KD, Kagiannas AG, Psarras J (2008) Energy policy making: an old concept or a modern challenge? Energy Sources Part B Econ Plan Policy 3(4):362–371. https://doi.org/10.1080/15567240701232378

    Article  Google Scholar 

  • Fisher FM (1961) The stability of the cournot oligopoly solution: the effects of speeds of adjustment and increasing marginal costs. Rev Econ Stud 28(2):125–135

    Article  Google Scholar 

  • Friedman J (1983) Oligopoly theory. Cambridge surveys of economic literature. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Furth D (1986) Stability and instability in oligopoly. J Econ Theory 40(2):197–228

    Article  Google Scholar 

  • Gabszewicz JJ, Vial JP (1972) Oligopoly “a la cournot” in a general equilibrium analysis. J Econ Theory 4(3):381–400

    Article  Google Scholar 

  • Hahn FH (1962) The stability of the cournot oligopoly solution. Rev Econ Stud 29(4):329–331

    Article  Google Scholar 

  • Hommes CH, Ochea MI, Tuinstra J (2018) Evolutionary competition between adjustment processes in cournot oligopoly: instability and complex dynamics. Dyn Games Appl 8(4):822–843

    Article  Google Scholar 

  • Huck S, Normann HT, Oechssler J (2002) Stability of the cournot process-experimental evidence. Int J Game Theory 31(1):123–136

    Article  Google Scholar 

  • Jury EI (1973) Theory and application of the z-transform method. Krieger Pub Co., Malabar

    Google Scholar 

  • Kakeya S (1912) On the limits of the roots of an algebraic equation with positive coefficients. Tohoku Math J First Ser 2:140–142

    Google Scholar 

  • Khalil H (2002) Nonlinear systems. Pearson Education, Prentice Hall, Upper Saddle River

    Google Scholar 

  • Kordonis I, Papavassilopoulos GP (2015) Lq nash games with random entrance: an infinite horizon major player and minor players of finite horizons. IEEE Trans Autom Control 60(6):1486–1500

    Article  Google Scholar 

  • Kordonis I, Charalampidis AC, Papavassilopoulos GP (2018) Pretending in dynamic games, alternative outcomes and application to electricity markets. Dyn Games Appl 8:1–30

    Article  Google Scholar 

  • MacCluer CR (2000) The many proofs and applications of Perron’s theorem. Siam Rev 42(3):487–498

    Article  Google Scholar 

  • McManus M, Quandt RE (1961) Comments on the stability of the cournot oligipoly model. Rev Econ Stud 28(2):136–139

    Article  Google Scholar 

  • Mickens R (1991) Difference equations, 2nd edn. Taylor & Francis, Abingdon-on-Thames

    Google Scholar 

  • Okuguchi K (1970) Adaptive expectations in an oligopoly model. Rev Econ Stud 37(2):233–237

    Article  Google Scholar 

  • Papavassilopoulos GP (1986) Iterative techniques for the nash solution in quadratic games with unknown parameters. SIAM J Control Optim 24(4):821–834

    Article  Google Scholar 

  • Papavassilopoulos GP (1988) Adaptive games. In: Albeverio S, Blanchard P, Hazewinkel M, Streit L (eds) Stochastic processes in physics and engineering. Mathematics and its applications, vol 42. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2893-0_13

  • Puu T (2008) On the stability of cournot equilibrium when the number of competitors increases. J Econ Behav Org 66(3–4):445–456

    Article  Google Scholar 

  • Puu T, Sushko I (2002) Oligopoly dynamics: models and tools. Springer, Berlin

    Book  Google Scholar 

  • Rassenti S, Reynolds SS, Smith VL, Szidarovszky F (2000) Adaptation and convergence of behavior in repeated experimental cournot games. J Econ Behav Org 41(2):117–146

    Article  Google Scholar 

  • Rumchev V, James D (1990) The role of non-negative matrices in discrete-time mathematical modelling. Int J Math Educ Sci Technol 21(2):169–182

    Article  Google Scholar 

  • Stamtsis G, Doukas H (2018) Cooperation or localization in european capacity markets? A coalitional game over graph approach. Energies. https://doi.org/10.3390/en11061473

    Article  Google Scholar 

  • Szidarovszky F, Okuguchi K (1988) A linear oligopoly model with adaptive expectations: stability reconsidered. J Econ 48(1):79–82

    Article  Google Scholar 

  • Theocharis RD (1960) On the stability of the cournot solution on the oligopoly problem. Rev Econ Stud 27(2):133–134

    Article  Google Scholar 

  • Thorlund-Petersen L (1988) Iterative computation of cournot equilibrium. Games Econ Behav 2:61–75. https://doi.org/10.1016/0899-8256(90)90014-L

    Article  Google Scholar 

  • von Mouche P, Quartieri F (2016) Equilibrium theory for Cournot oligopolies and related games. Springer, Berlin

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the participants of the 10th Workshop in Dynamic Games in Management Science (DGMS2018), November 1–2, 2018, Morocco for their comments, which improved this work.

Author information

Authors and Affiliations

  1. Research Centre for Systems & Control, School of Mathematics, Computer Science and Engineering, City, University of London, Northampton Square, London, EC1V 0HB, UK

    Nikolaos Chrysanthopoulos

  2. School of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytexneio Str., Zografou, 15773, Greece

    George P. Papavassilopoulos

Authors
  1. Nikolaos Chrysanthopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. George P. Papavassilopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolaos Chrysanthopoulos.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chrysanthopoulos, N., Papavassilopoulos, G.P. Adaptive rules for discrete-time Cournot games of high competition level markets. Oper Res Int J 21, 2879–2906 (2021). https://doi.org/10.1007/s12351-019-00522-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12351-019-00522-z

Keywords

Mathematics Subject Classification

Search

Navigation