Mixed Strategies for Hierarchical Zero-Sum Games

  • Lina Mallozzi
  • Jacqueline Morgan
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 6)


New concepts of approximate mixed solutions for hierarchical saddle-point problems are introduced in the situation in which the leader cannot influence the followers and minimize the worst, under the lack of convexity assumptions. Then sufficient conditions for the existence of such approximate solutions and convergence of the corresponding values are presented.


Variational Inequality Saddle Point Mixed Strategy Convexity Assumption Linearize Objective Function 
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  1. [1]
    Aubin, J. P.Mathematical Methods of Game and Economic TheoryNorth-Holland, Amsterdam, 1982.Google Scholar
  2. [2]
    Basar, T. and Olsder, G.J. Dynamic Noncooperative GamesAcademic Press, New York, 1995.Google Scholar
  3. [3]
    Borel, E. La theorie du jeu et les equations integrales a noyeau symmetriqueComptes Rendus Aca. Sc., 173, pp. 1304–1308, 1921.MATHGoogle Scholar
  4. [4]
    Breton, M., Alj, A., and Haurie, A. Sequential Stackelberg equilibria in two-person gamesJ. Opt. Th. Appl.59, pp. 71–97, 1988.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Cruz, C. and Cruz, J. Stackelberg solution for two-person games with biased information patternsIEEE Trans. Auto. Cont. AC-17, pp. 791–798, 1972.MATHCrossRefGoogle Scholar
  6. [6]
    Harker, P. T. and Pang, J. S. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applicationsMath. Program.48, pp. 161–220, 1990.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Leitmann, G. On Generalized Stackelberg StrategiesJ. Opt. Th. Appl. 26pp. 637–6431978.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Luo, Z. Q., Pang, J. S., and Ralph, D.Mathematical Programs with Equilibrium ConstraintsCambridge Univ. Press, New York, 1996.CrossRefGoogle Scholar
  9. [9]
    Mallozzi, L. and Morgan, J.s-mixed strategies for static continuous kernel stackelberg problemJ. Opt. Th. Appl., 78, pp. 303–316, 1993.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Mallozzi, L. and Morgan, J. Weak Stackelberg problem and mixed solutions under data perturbationsOptimization32, pp. 269–290, 1995.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Mallozzi, L. and Morgan, J. Hierarchical systems with weighted reaction setin Nonlinear Optimization and Applications D. Pillo and F. Giannessi, eds. Plenum Press New Yorkpp. 271–2831996.Google Scholar
  12. [12]
    Morgan, J. and Raucci, R. Continuity properties of s-solutions for generalized parametric saddle point problems and application to hierarchical gamesJ. Math. Anal. Appl.211, pp. 30–48, 1997a.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Morgan, J. and Raucci, R. Approximate solutions for strong hierarchical saddle point problemsAtti Sem. Mat. Fis. Univ. Modena, XLV, pp. 395–409, 1997b.Google Scholar
  14. [14]
    Moulin, H.Game Theory for the Social SciencesNYU Press, New York, 1986.Google Scholar
  15. [15]
    Owen, G.Game TheoryAcademic Press, New York, 1982.MATHGoogle Scholar
  16. [16]
    Schwartz, L. Analyse III Calcul IntegralHermann, Paris, 1993.Google Scholar
  17. [17]
    Sheraly, H. D., Soyster, A. L., and Murphy, F. H.Stackelberg-Nash-Cournot equilibria: Characterizations and computationsOper. Res., 31, pp. 253–276, 1983.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Simaan, M. and Cruz, J. On the Stackelberg strategies in nonzero sum gamesJ. Opt. Th. Appl.11, pp. 533–555, 1973.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Tobin, R. L. Uniqueness results and algorithm for Stackelberg-Cournot-Nash equilibriaAnn. Oper. Res.34, pp. 21–36, 1992.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    von NeumannJ.and Morgenstern, O.Theory of Games and Economic Behavior, Princeton Univ. Press, Princeton, 1934.Google Scholar
  21. [21]
    von Stackelberg, H. Marktform und GleichgewichtJulius Springer, Vienna, 1934.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Lina Mallozzi
    • 1
  • Jacqueline Morgan
    • 1
  1. 1.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Napoli “Federico II” NapoliItaly

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