Cournot Games with Limited Demand: From Multiple Equilibria to Stochastic Equilibrium

  • Ido Polak
  • Nicolas Privault


We construct Cournot games with limited demand, resulting into capped sales volumes according to the respective production shares of the players. We show that such games admit three distinct equilibrium regimes, including an intermediate regime that allows for a range of possible equilibria. When information on demand is modeled by a delayed diffusion process, we also show that this intermediate regime collapses to a single equilibrium while the other regimes approximate the deterministic setting as the delay tends to zero. Moreover, as the delay approaches zero, the unique equilibrium achieved in the stochastic case provides a way to select a natural equilibrium within the range observed in the no lag setting. Numerical illustrations are presented when demand is modeled by an Ornstein–Uhlenbeck process and price is an affine function of output.


Game theory Multiple equilibria Equilibrium selection Limited demand Delayed information Stochastic control 

Mathematics Subject Classification

91A05 91A10 91A18 91B24 



This research was supported by the Singapore MOE Tier 2 Grant MOE2016-T2-1-036.


  1. 1.
    Baghery, F., Øksendal, B.: A maximum principle for stochastic control with partial information. Stoch. Anal. Appl. 25(3), 705–717 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Caraballo, M.A., Mármol, A.M., Monroy, L., Buitrago, E.M.: Cournot competition under uncertainty: conservative and optimistic equilibria. Rev. Econ. Design 19, 145–165 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lagerlöf, J.N.M.: Equilibrium uniqueness in a Cournot model with demand uncertainty. Topics Theor. Econ. 6(1), 1–6 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Lagerlöf, J.N.M.: Insisting on a non-negative price: oligopoly, uncertainty, welfare, and multiple equilibria. Int. J. Ind. Organ. 25, 861–875 (2007)CrossRefGoogle Scholar
  5. 5.
    Øksendal, B., Sandal, L., Ubøe, J.: Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models. J. Econ. Dyn. Control 37(7), 1284–1299 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Øksendal, B., Sulem, A.: Forward-backward stochastic differential games and stochastic control under model uncertainty. J. Optim. Theory Appl. 161(1), 22–55 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Menoukeu Pamen, O.: Optimal control for stochastic delay systems under model uncertainty: A stochastic differential game approach. J. Optim. Theory Appl. 167, 998–1031 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Polak, I., Privault, N.: A stochastic newsvendor game with dynamic retail prices. J. Ind. Manag. Optim. (2018, to appear)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

Personalised recommendations