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Mathematical Programming

, Volume 104, Issue 2–3, pp 407–435 | Cite as

Collusive game solutions via optimization

  • J.E. Harrington
  • B.F. Hobbs
  • J.S. PangEmail author
  • A. Liu
  • G. Roch
Article

Abstract

A Nash-based collusive game among a finite set of players is one in which the players coordinate in order for each to gain higher payoffs than those prescribed by the Nash equilibrium solution. In this paper, we study the optimization problem of such a collusive game in which the players collectively maximize the Nash bargaining objective subject to a set of incentive compatibility constraints. We present a smooth reformulation of this optimization problem in terms of a nonlinear complementarity problem. We establish the convexity of the optimization problem in the case where each player's strategy set is unidimensional. In the multivariate case, we propose upper and lower bounding procedures for the collusive optimization problem and establish convergence properties of these procedures. Computational results with these procedures for solving some test problems are reported.

Mathematics Subject Classification (1991)

90C26 90C33 90C90 91A10 91A20 

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References

  1. 1.
    Bernheim, B.D., Whinston, M.D.: Multimarket contact and collusive behavior. RAND J. Economics 21, 1–26 (1990)Google Scholar
  2. 2.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer-Verlag, New York, 2000Google Scholar
  3. 3.
    Bunn, D.W., Oliveira, F.S.: Evaluating individual market power in electricity markets via agent-based simulation. Ann. Oper. Res. 121, 57–77 (2003)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Danskin, J.M.: The theory of min-max with applications. SIAM J. Appl. Math. 14, 641–664 (1966)CrossRefGoogle Scholar
  5. 5.
    Dirkse, S.P., Ferris, M.C.: The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems. Optimization Methods and Software 5, 123–156 (1995)Google Scholar
  6. 6.
    Dubey, P.: Inefficiency of Nash equilibria. Math. Oper. Res. 11, 1–8 (1986)Google Scholar
  7. 7.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer-Verlag, New York, 2003Google Scholar
  8. 8.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. Duxbury Press, Brooks/Cole Publishing Company, 2002Google Scholar
  9. 9.
    Friedman, J.W.: Oligopoly and the Theory of Games. North-Holland, Amsterdam, 1977Google Scholar
  10. 10.
    Fudenberg, D., Tirole, J.: Game Theory. MIT Press, Cambridge, 1991Google Scholar
  11. 11.
    Harrington, J.E., Jr.: Collusion in multiproduct oligopoly games under a finite horizon. International Economic Review 28, 1–14 (1987)Google Scholar
  12. 12.
    Harrington, J.E., Jr.: The determination of price and output quotas in a heterogeneous cartel. International Economic Review 32, 767–792 (1991)Google Scholar
  13. 13.
    Herrero, M.J.: The Nash program: non-convex bargaining problems. J. Economic Theory 49, 266–277 (1989)CrossRefGoogle Scholar
  14. 14.
    Horst, R., Thoai, N.V.: DC programming: Overview. J. Optimization Theory and Applications 103, 1–43 (1999)CrossRefGoogle Scholar
  15. 15.
    Kaneko, M.: An extension of the Nash bargaining problem and the Nash social welfare function. Theory and Decision 12, 135–148 (1980)CrossRefGoogle Scholar
  16. 16.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs With Equilibrium Constraints. Cambridge University Press, Cambridge, England, 1996Google Scholar
  17. 17.
    Nash, J.F.: The bargaining problem. Econometrica 28, 155–162 (1950)Google Scholar
  18. 18.
    Osborne, M.J., Rubinstein, A.: Bargaining and Markets. Academic Press, San Diego, 1990Google Scholar
  19. 19.
    Puller, S.L.: Pricing and firm conduct in California's deregulated electricity market. PWP-080, Power Program, University of California Energy Institute, Berkeley, 2001Google Scholar
  20. 20.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer-Verlag, Berlin, 1998Google Scholar
  21. 21.
    Rothkopf, M.H.: Daily repetition: A neglected factor in the analysis of electricity auctions. The Electricity J. 12, 61–70 (1999)Google Scholar
  22. 22.
    Schmalensee, R.: Competitive advantage and collusive optima. International J. Industrial Organization 5, 351–367 (1987)CrossRefGoogle Scholar
  23. 23.
    Sweeting, A.: Market outcomes and generator behaviour in the England and Wales wholesale electricity market, 1995-2000. Department of Economics, Massachusetts Insitute of Technology, presented at the 7th Annual POWER Research Conference on Electricity Industry Restructuring, University of California Energy Institute, Berkeley, March 22, 2002Google Scholar
  24. 24.
    Tao, P.D., An, L.T.H.: A d.c. optimization algorithm for solving the trust-region subproblem. SIAM Journal on Optimization 8, 476–505 (1998)Google Scholar
  25. 25.
    Tawarmalani, M., Sahinidis, N.V.: Convex extensions and envelopes of lower semi-continuous functions. Mathematical Programming, Series A 93, 247–263 (2002)Google Scholar
  26. 26.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. (Kluwer Academic Publishers, Dordrecht, 2002). [Volume 65 in “Nonconvex Optimization And Its Applications” series.]Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • J.E. Harrington
    • 1
  • B.F. Hobbs
    • 2
  • J.S. Pang
    • 3
    Email author
  • A. Liu
    • 4
  • G. Roch
    • 5
  1. 1.Department of EconomicsThe Johns Hopkins UniversityMarylandUSA
  2. 2.Department of Geography and Environmental EngineeringThe Johns Hopkins UniversityMarylandUSA
  3. 3.Department of Mathematical SciencesRensselaer Polytechnic InstituteNew YorkUSA
  4. 4.Department of Applied Mathematics and StatisticsThe Johns Hopkins UniversityMarylandUSA
  5. 5.Department of Applied Mathematics and StatisticsThe Johns Hopkins UniversityMarylandUSA

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