The Annals of Regional Science

, Volume 28, Issue 3, pp 263–283 | Cite as

A dynamical systems approach for network oligopolies and variational inequalities

  • Anna Nagurney
  • Paul Dupuis
  • Ding Zhang


The variational inequality problem has been used to formulate and study a plethora of competitive equilibrium problems at the equilibrium state. In this paper, we focus on oligopolistic market network equilibrium problems in which firms are spatially located and seek to determine their profit-maximizing production out-puts and shipments, in the presence of transportation costs. In particular, we utilize the equivalence between the set of stationary points of a dynamical system and the set of solutions to the associated variational inequality problem governing the network oligopoly problem to explore the underlying dynamics both qualitatively and numerically. Although the dynamical system is nonstandard in that the right-hand side is discontinuous, recent theoretical results have shown that the important qualitative and quantitative results of ordinary differential equations are applicable under the standard Lipschitz continuity assumptions. The identification between solutions to dynamical systems and associated variational inequality problems unveils a new tool for addressing the behavior of competitive network systems over time.


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  1. 1.
    Arrow, K. J., Hurwicz, L. (1958) “Gradient methods for concave programming, I local results.” In Arrow, K. J., Hurwicz, L., Uzawa, H. (eds.) Studies in linear and nonlinear programming, Stanford University Press, Stanford, California, pp 117–126.Google Scholar
  2. 2.
    Beckmann, M., McGuire, C. B., Winsten, C. B. (1956) Studies in the economics of transportation. Yale University Press, New Haven, Connecticut.Google Scholar
  3. 3.
    Bertsekas, D. P. (1976) “On the Goldstein-Levitin-Polyak gradient projection method.”IEEE Transactions on Automatic Control AC-21: 174–184.Google Scholar
  4. 4.
    Cournot, A. A. (1838) Researches into the mathematical principles of the theory of wealth. English translation, MacMillan, London, England, 1897.Google Scholar
  5. 5.
    Dafermos, S. (1980) “Traffic equilibrium and variational inequalities.”Transportation Science 14: 42–54.Google Scholar
  6. 6.
    Dafermos, S. (1982) “The general multimodal traffic equilibrium problem with elastic demand.”Networks 12: 57–72.Google Scholar
  7. 7.
    Dafermos, S. (1983) “An iterative scheme for variational inequalities.”Mathematical Programming 26: 40–47.Google Scholar
  8. 8.
    Dafermos, S., Nagurney, A. (1987) “Oligopolistic and competitive behavior of spatially separated markets.”Regional Science and Urban Economics 17: 245–254.Google Scholar
  9. 9.
    Dupuis, P. (1987) “Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets.”Stochastics 21: 63–96.Google Scholar
  10. 10.
    Dupuis, P., Ishii, H. (1991) “On Lipschitz continuity of the solution mapping to the Skorokhod Problem, with applications.”Stochastics and Stochastic Reports 35: 31–62.Google Scholar
  11. 11.
    Dupuis, P., Nagurney, A. (1993) “Dynamical systems and variational inequalities.”Annals of Operations Research 44, no.3: 9–42.Google Scholar
  12. 12.
    Dupuis, P., Williams, R. J. (1992) “Lyapunov functions for semimartingale reflected Brownian motions.” Lefschetz Center for Dynamical Systems, Report # 92-5, Brown University, Providence, Rhode Island.Google Scholar
  13. 13.
    Fillipov, A. N. (1960) “Differential equations with discontinuous right-hand side.”Mat. Sbornik (N. S.)51: 99–128.Google Scholar
  14. 14.
    Flam, S. P., Ben-Israel, A. (1990) “A continuous approach to oligopolistic market equilibrium.”Operations Research 38: 1045–1051.Google Scholar
  15. 15.
    Florian, M., Los, M. (1982) “A new look at static spatial price equilibrium problems.”Regional Science and Urban Economics 12: 579–597.Google Scholar
  16. 16.
    Gabay, D., Moulin, H. (1980) “On the uniqueness and stability of Nash equilibria in noncooperative games.” In: Bensoussan, A., Kleindorfer, P., Tapiero, C. S. (eds.) Applied stochastic control of econometrics and management science. North-Holland, Amsterdam, The Netherlands.Google Scholar
  17. 17.
    Goldstein, A. A. (1964) “Convex programming in Hilbert space.”Bulletin of the American Mathematical Society 70: 709–710.Google Scholar
  18. 18.
    Goldstein, A. A. (1967) Constructive real analysis. Harper & Row, New York.Google Scholar
  19. 19.
    Hartman, P., Stampacchia, G. (1966) “On some nonlinear elliptic differential functional equations,”Acta Mathematica 115: 271–310.Google Scholar
  20. 20.
    Haurie, A., Marcotte, P. (1985) “On the relationship between Nash-Cournot and Wardrop equilibria.”Networks 15: 295–308.Google Scholar
  21. 21.
    Hildreth, C. (1957) “A quadratic programming procedure.”Naval Research Logistics Quarterly 4: 79–85.Google Scholar
  22. 22.
    Kinderlehrer, D., Stampacchia G. (1980) An introduction to variational inequalities and their applications. Academic Press, New York.Google Scholar
  23. 23.
    Knight, F. H. (1924) “Some fallacies in the interpretation of social costs.”Quarterly Journal of Economics 38: 582–606.Google Scholar
  24. 24.
    Levitin, E. S., Polyak, B. T. (1966) “Constrained minimization problems.”USSR Comput. Math. Math. Phys. 6: 1–50.Google Scholar
  25. 25.
    Murphy, F. H., Sherali, H. D., Soyster, A. L. (1982) “A mathematical programming approach for determining oligopolistic market equilibrium.”Mathematical Programming 24: 92–106.Google Scholar
  26. 26.
    Nagurney, A. (1988) “Algorithms for oligopolistic market equilibrium problems.”Regional Science and Urban Economics 18: 425–445.Google Scholar
  27. 27.
    Nagurney, A. (1993) Network economics: a variational inequality approach. Kluwer Academic Publishers, Boston, Massachusetts.Google Scholar
  28. 28.
    Nagurney, A., Dong, J., Hughes, M. (1992) “The formulation and computation of general financial equilibrium.”Optimization 26: 339–354.Google Scholar
  29. 29.
    Okuguchi, K. (1976) Expectations and stability in oligopoly models.Lecture notes in economics and mathematical systems 138. Springer-Verlag, Berlin, Germany.Google Scholar
  30. 30.
    Okuguchi, K., Szidarovsky, F. (1990) The theory of oligopoly with multi-product firms.Lecture notes in economics and mathematical systems 342. Springer-Verlag, Berlin, Germany.Google Scholar
  31. 31.
    Pigou, A. C. (1920) The economics of welfare. MacMillan Company, London, England.Google Scholar
  32. 32.
    Qiu, Y. (1991) “Solution properties of oligopolistic network equilibria.”Networks 71: 565–580.Google Scholar
  33. 33.
    Samuelson, P. A. (1952) “Spatial price equilibrium and linear programming.”American Economic Review 42: 283–303.Google Scholar
  34. 34.
    Skorokhod, A. V. (1961) “Stochastic equations for diffusions in a bounded region.”Theory of Probability and its Applications 6: 264–274.Google Scholar
  35. 35.
    Takayama, T., Judge, G. G. (1971) Spatial and temporal price and allocation models. North-Holland, Amsterdam, the Netherlands.Google Scholar
  36. 36.
    Zhao, L., Nagurney, A. (1993) “A network framework for general economic equilibrium.” In Du, D., Pardalos, P. M. (eds.) Network optimization problems: algorithms, complexity, and applications, World Scientific Press, Singapore, pp. 363–386.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Anna Nagurney
    • 1
  • Paul Dupuis
    • 2
  • Ding Zhang
    • 3
  1. 1.School of ManagementUniversity of MassachusettsAmherstUSA
  2. 2.Lefschetz Center for Dynamical Systems, Division of Applied MathematicsBrown UniversityProvidenceUSA
  3. 3.Department of Industrial Engineering and Operations ResearchUniversity of MassachusettsAmherstUSA

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