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

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Abstract

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.

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References

  1. NASH, J. F., Equilibrium Points in n-Person Games, Proceedings of the National Academy of Sciences, Vol. 36, pp. 48–49, 1950.

    Article  MathSciNet  MATH  Google Scholar 

  2. MILLHAM, C. B., On Nash Subsets of Bimatrix Games, Naval Research Logistics Quarterly, Vol. 74, pp. 307–317, 1974.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  4. MILLS, H., Equilibrium Points in Finite Games, SIAM Journal on Applied Mathematics, Vol. 8, pp. 397–402, 1960.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  6. MANGASARIAN, O. L., Equilibrium Points of Bimatrix Games, SIAM Journal on Applied Mathematics, Vol. 12, pp. 778–780, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  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].

  8. KEIDING, H., On the Maximal Number of Nash Equilibria in a Bimatrix Game, Games and Economic Behavior, Vol. 21, pp. 148–160, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  9. VON STENGEL, B., New Maximal Numbers of Equilibria in Bimatrix Games, Discrete and Computational Geometry, Vol. 21, pp. 557–568, 1998.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  11. LEMKE, C. E., and HOWSON, T. T., Equilibrium Points of Bimatrix Games, SIAM Journal on Applied Mathematics, Vol. 12, pp. 413–423, 1961.

    Article  MathSciNet  Google Scholar 

  12. DICKHAUT, J., and KAPLAN, T., A Program for Finding Nash Equilibria, Mathematica Journal, Vol. 1, pp. 87–93, 1991.

    Google Scholar 

  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.

  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.

  15. AUDET, C., Optimisation Globale Structurée: Propriétés, Equivalences et Résolution, PhD Thesis, École Polytechnique de Montréal, pp. 91–109, 1997.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  18. HOWSON, J. T., Equilibria of Polymatrix Games, Management Science, Vol. 18, pp. 312–318, 1972.

    MathSciNet  MATH  Google Scholar 

  19. QUINTAS, L. G., A Note on Polymatrix Games, International Journal of Game Theory, Vols. 18–19, pp. 261–272, 1989.

  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.

  21. YANOVSKAYA, E. B., Equilibrium Points in Polymatrix Games, Latvian Mathematical Collection, Vol. 8, pp. 381–384, 1968.

    Google Scholar 

  22. COTTLE, R. W., and DANTZIG, G. B., Linear Programming Extensions, Princeton University Press, Princeton, New Jersey, 1963.

    Google Scholar 

  23. LEMKE, C. E., Bimatrix Games Equilibrium Points and Mathematical Programming, Management Science, Vol. 11, pp.xxx-xxx, 1965.

    MathSciNet  Google Scholar 

  24. EAVES, C. B., Polymatrix Games with Joint Constraints, SIAM Journal on Applied Mathematics, Vol. 24, pp. 418–423, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  26. MYERSON, R. B., Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, Massachusetts, 1997.

    Google Scholar 

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Authors and Affiliations

  1. GERAD and École Polytechnique de Montréal, Montréal, Québec, Canada

    C. Audet (Professor)

  2. École Polytechnique de Montréal, Montréal, Québec, Canada

    S. Belhaiza (PhD Student)

  3. GERAD and HEC Montréal, Montréal, Québec, Canada

    P. Hansen (Professor)

Authors
  1. C. Audet
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  2. S. Belhaiza
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  3. P. Hansen
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Additional information

Communicated by P. M. Pardalos

The authors thank Bernhard von Stengel for constructive comments on the contents of this paper.

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Audet, C., Belhaiza, S. & Hansen, P. Enumeration of All the Extreme Equilibria in Game Theory: Bimatrix and Polymatrix Games. J Optim Theory Appl 129, 349–372 (2006). https://doi.org/10.1007/s10957-006-9070-3

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  • DOI: https://doi.org/10.1007/s10957-006-9070-3

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