Enumeration of All the Extreme Equilibria in Game Theory: Bimatrix and Polymatrix Games

  • C. Audet
  • S. Belhaiza
  • P. Hansen


Bimatrix and polymatrix games are expressed as parametric linear 0–1 programs. This leads to an algorithm for the complete enumeration of their extreme equilibria, which is the first one proposed for polymatrix games. The algorithm computational experience is reported for two and three players on randomly generated games for sizes up to 14 × 14 and 13 × 13 × 13.


Bimatrix games polymatrix games Nash equilibria extreme equilibria enumeration 


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  1. 1.
    NASH, J. F., Equilibrium Points in n-Person Games, Proceedings of the National Academy of Sciences, Vol. 36, pp. 48–49, 1950.CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    MILLHAM, C. B., On Nash Subsets of Bimatrix Games, Naval Research Logistics Quarterly, Vol. 74, pp. 307–317, 1974.MathSciNetGoogle Scholar
  3. 3.
    AUDET, C., HANSEN, P., JAUMARD, B., and SAVARD, G., Enumeration of All Extreme Equilibrium Strategies of Bimatrix Games, SIAM Journal on Statistical and Scientific Computing, Vol. 23, pp. 323–338, 2001.CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    MILLS, H., Equilibrium Points in Finite Games, SIAM Journal on Applied Mathematics, Vol. 8, pp. 397–402, 1960.CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    MANGASARIAN, O. L., and STONE, H., Two-Person Nonzero-Sum Games and Quadratic Programming, Journal of Mathematical Analysis and Applications, Vol. 9, pp. 348–355, 1964.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    MANGASARIAN, O. L., Equilibrium Points of Bimatrix Games, SIAM Journal on Applied Mathematics, Vol. 12, pp. 778–780, 1964.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    VOROBEV, N. N., Equilibrium Points in Bimatrix Games, Theoriya Veroyatnostej i ee Primwneniya, Vol. 3, pp. 318–331, 1958 [English Version: Theory of Probability and Its Applications, Vol. 3, pp. 297–309, 1958].Google Scholar
  8. 8.
    KEIDING, H., On the Maximal Number of Nash Equilibria in a Bimatrix Game, Games and Economic Behavior, Vol. 21, pp. 148–160, 1997.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    VON STENGEL, B., New Maximal Numbers of Equilibria in Bimatrix Games, Discrete and Computational Geometry, Vol. 21, pp. 557–568, 1998.MathSciNetGoogle Scholar
  10. 10.
    KUHN, H. W., An Algorithm for Equilibrium Points in Bimatrix Games, Proceedings of the National Academy of Sciences, Vol. 47, pp. 1657–1662, 1961.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    LEMKE, C. E., and HOWSON, T. T., Equilibrium Points of Bimatrix Games, SIAM Journal on Applied Mathematics, Vol. 12, pp. 413–423, 1961.CrossRefMathSciNetGoogle Scholar
  12. 12.
    DICKHAUT, J., and KAPLAN, T., A Program for Finding Nash Equilibria, Mathematica Journal, Vol. 1, pp. 87–93, 1991.Google Scholar
  13. 13.
    MCKELVEY, R. D., and MCLENNAN, A., Computation of Equilibria in Finite Games, Handbook of Computational Economics, Edited by H. M. Amman, D. A. Kendrick, and J. Rust, Elsevier, Amsterdam, Holland, Vol. 1, pp. 87–142, 1996.Google Scholar
  14. 14.
    WINKELS, R., An Algorithm to Determine all Equilibrium Points of a Bimatrix Games, Game Theory and Related Topics, Edited by O. Moeschlin and D. Pallaschke, North Holland Publishing Company, Amsterdam, Holland, 1972.Google Scholar
  15. 15.
    AUDET, C., Optimisation Globale Structurée: Propriétés, Equivalences et Résolution, PhD Thesis, École Polytechnique de Montréal, pp. 91–109, 1997.Google Scholar
  16. 16.
    JÚDICE, J., and MITRA, G., Reformulations of Mathematical Programming Problems as Linear Complementarity Problems and Investigation of Their Solution Methods, Journal of Optimization Theory and Applications, Vol. 57, pp. 123–149, 1988.CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    AUDET, C., HANSEN, P., JAUMARD, B., and SAVARD, G., Links between Linear Bilevel and Mixed 0-1 Programming Problems, Journal of Optimization Theory and Applications, Vol. 93, pp. 273–300, 1997.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    HOWSON, J. T., Equilibria of Polymatrix Games, Management Science, Vol. 18, pp. 312–318, 1972.MathSciNetzbMATHGoogle Scholar
  19. 19.
    QUINTAS, L. G., A Note on Polymatrix Games, International Journal of Game Theory, Vols. 18–19, pp. 261–272, 1989.Google Scholar
  20. 20.
    COTTLE, R. W., and DANTZIG, G. B., Complementary Pivot Theory of Mathematical Programming, Mathematics of the Decision Scienes, Part 1, Vol. 11, pp. xxx–xxx, 1968.Google Scholar
  21. 21.
    YANOVSKAYA, E. B., Equilibrium Points in Polymatrix Games, Latvian Mathematical Collection, Vol. 8, pp. 381–384, 1968.Google Scholar
  22. 22.
    COTTLE, R. W., and DANTZIG, G. B., Linear Programming Extensions, Princeton University Press, Princeton, New Jersey, 1963.Google Scholar
  23. 23.
    LEMKE, C. E., Bimatrix Games Equilibrium Points and Mathematical Programming, Management Science, Vol. 11,, 1965.MathSciNetGoogle Scholar
  24. 24.
    EAVES, C. B., Polymatrix Games with Joint Constraints, SIAM Journal on Applied Mathematics, Vol. 24, pp. 418–423, 1973.CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    HOWSON, J. T., and ROSENTHAL, R. W., Bayesian Equilibria of Finite Two-Person Games with Incomplete Information, Management Science, Vol. 21, pp. 313–315, 1974.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    MYERSON, R. B., Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, Massachusetts, 1997.Google Scholar

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© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • C. Audet
    • 1
  • S. Belhaiza
    • 2
  • P. Hansen
    • 3
  1. 1.GERAD and École Polytechnique de MontréalMontréalCanada
  2. 2.École Polytechnique de MontréalMontréalCanada
  3. 3.GERAD and HEC MontréalMontréalCanada

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