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On Cournot-Nash-Walras Equilibria and Their Computation


This paper concerns a model of Cournot-Nash-Walras (CNW) equilibrium where the Cournot-Nash concept is used to capture equilibrium of an oligopolistic market with non-cooperative players/firms who share a certain amount of a so-called rare resource needed for their production, and the Walras equilibrium determines the price of that rare resource. We prove the existence of CNW equilibria under reasonable conditions and examine their local stability with respect to small perturbations of problem data. In this way we show the uniqueness of CNW equilibria under mild additional requirements. Finally, we suggest some efficient numerical approaches and compute several instances of an illustrative test example.

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  1. Britz, W., Ferris, M.C., Kuhn, A.: Modeling water allocating institutions based on multiple Optimization problems with equilibrium constraints. Environ. Model. Softw. 46, 196–207 (2013)

    Article  Google Scholar 

  2. Brooke, A., Kendrick, D., Meeraus, A.: GAMS: a User’s Guide. The Scientific Press, South San Francisco (1988)

    Google Scholar 

  3. Cao, M., Ferris, M.C.: A pivotal method for affine variational inequalities. Math. Oper. Res. 21, 44–64 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  4. Dirkse, S.P., Ferris, M.C.: The PATH solver: a Non-Monotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5, 123–156 (1995)

    Article  Google Scholar 

  5. Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 7, 1087–1105 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  6. Dontchev, A.L., Rockafellar, R.T.: Implicit functions and solution mappings. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  7. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalaities and Complementarity Problems. Springer, New York (2003)

    MATH  Google Scholar 

  8. Ferris, M.C., Dirkse, S.P., Jagla, J.-H., Meeraus, A.: An extended mathematical programming framework. Comput. Chem. Eng. 33, 1973–1982 (2009)

    Article  Google Scholar 

  9. Ferris, M.C., Munson, T.S.: Interfaces to PATH 3.0: design, implementation and usage. Comput. Optim. Appl. 12, 207–227 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  10. Flåm, S.D.: Noncooperative games, coupling constraints and partial efficiency. Econ. Theory Bull., 1–17 published electronically (2016)

  11. Flåm, S.D., Antipin, A.S.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  12. Kanzow, C.H., Schwartz, A.: The price of inexactness: convergence properties of relaxation methods for mathematical programs with equilibrium constraints revisited. Math. Oper. Res. 40, 253–275 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  13. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation Basic Theory, vol. 1. Springer, Berlin (2006)

    Google Scholar 

  14. Murphy, F.H., Sherali, H.D., Soyster, A.L.: A mathematical programming approach for determining oligopolistic market equilibrium. Math. Program. 24, 92–106 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  15. Ortega, J.M., Rheinbolt, W.C.: Iterative Solutions of Nonlinear Equations in Several Variables. Acad Press, New York (1970)

    Google Scholar 

  16. Outrata, J.V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  17. Outrata, J.V., Červinka, M.: On the implicit programming approach in a class of mathematical programs with equilibrium constraints. Control. Cybern. 38, 1557–1574 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–299 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  19. Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  20. Rosen, J.B.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33, 520–534 (1965)

    MathSciNet  Article  MATH  Google Scholar 

  21. Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Jiří V. Outrata.

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This work was partially supported by the Grant Agency of the Czech Republic under Grant P402/12/1309 and 15-00735S, by the Grant Agency of the Charles University by Grant SFG 2567, by the Australian Research Council under grant DP-110102011 and in part by a grant from the Department of Energy and funding from the USDA.

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Outrata, J.V., Ferris, M.C., Červinka, M. et al. On Cournot-Nash-Walras Equilibria and Their Computation. Set-Valued Var. Anal 24, 387–402 (2016).

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  • Cournot-Nash-Walras equilibrium
  • Existence
  • Stationarity conditions
  • Stability

Mathematics Subject Classification (2010)

  • 90C33
  • 91B52
  • 49J40
  • 90C31