Practical and Efficient Approximations of Nash Equilibria for Win-Lose Games Based on Graph Spectra

  • Haralampos Tsaknakis
  • Paul G. Spirakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)

Abstract

It is shown here that the problem of computing a Nash equilibrium for two-person games can be polynomially reduced to an indefinite quadratic programming problem involving the spectrum of the adjacency matrix of a strongly connected directed graph on n vertices, where n is the total number of players’ strategies. Based on that, a new method is presented for computing approximate equilibria and it is shown that its complexity is a function of the average spectral energy of the underlying graph. The implications of the strong connectedness properties on the energy and on the complexity of the method is discussed and certain classes of graphs are described for which the method is a polynomial time approximation scheme (PTAS). The worst case complexity is bounded by a subexponential function in the total number of strategies n and a comparison is made with a previously reported method with subexponential complexity.

References

  1. 1.
    Abbot, T., Kane, D., Valiant, P.: On the Complexity of Two-Player Win-Lose Games. In: FOCS 2005 (2005)Google Scholar
  2. 2.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  3. 3.
    Bang-Jensen, J., Gutin, G.: Digraphs Theory, Algorithms and Applications. Springer, Heidelberg (2007)MATHGoogle Scholar
  4. 4.
    Bollobas, B.: Modern Graph Theory. Springer, New York (1998)CrossRefMATHGoogle Scholar
  5. 5.
    Chen, X., Deng, X., Teng, S.: Computing Nash Equilibria: Approximation and smoothed complexity. In: Proc. of the 47th IEEE Symp. on Foundations of Comp. Sci (FOCS 2006), pp. 603–612. IEEE Press, Sci (FOCS (2006)Google Scholar
  6. 6.
    Chen, X., Deng, X.: Settling the complexity of 2-player Nash equilibrium. In: Proc. of the 47th IEEE Symp. on Foundations of Comp. Sci. (FOCS 2006), pp. 261–272. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  7. 7.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The Complexity of Computing a Nash Equilibrium. In: Proceedings of STOC (2006)Google Scholar
  8. 8.
    Ebrahimi, J., Mohar, B., Nikiforov, V., Sheikn, A.: On the sum of two largest eigenvalues of a symmetric matrix. Linear Algebra and its Applications 429, 2781–2787 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kannan, R., Theobald, T.: Games of fixed rank: A hierarchy of bimatrix games. In: Proc. Symposium on Discrete Algorithms, New Orleans, LA (2007)Google Scholar
  10. 10.
    Lipton, R.J., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: EC 2003: Proceedings of the 4th ACM Conference on Electronic Commerce, pp. 36–41 (2003)Google Scholar
  11. 11.
    Nikiforov, V.: The energy of graphs and matrices. Journal of Mathematical Analysis and Applications 326, 1472–1475 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48(3), 498–532 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Tsaknakis, H., Spirakis, P.G.: An Optimization Approach for Approximate Nash Equilibria. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 42–56. Springer, Heidelberg (2007); Also, in ECCC TR07-067, 2nd revision (2007)Google Scholar
  14. 14.
    Tsaknakis, H., Spirakis, P.G.: An Optimization Approach for Approximate Nash Equilibria. Internet Mathematics 5(4), 363–382 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Tsaknakis, H., Spirakis, P.G., Kanoulas, D.: Performance Evaluation of a Descent Algorithm for Bi-matrix Games. In: Papadimitriou, C., Zhang, S. (eds.) WINE 2008. LNCS, vol. 5385, pp. 222–230. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Tsaknakis, H., Spirakis, P.G.: A Graph Spectral Approach for Computing Approximate Nash Equilibria. In: Electronic Colloquium on Computational Complexity (ECCC), TR09-096 (2009)Google Scholar
  17. 17.
    Ye, Y.: Computational Economy Equilibrium and its Application: Progresses on computing Arrow-Debreu-Leontief Competitive Equilibria. In: Papadimitriou, C., Zhang, S. (eds.) WINE 2008. LNCS, vol. 5385, p. 14. Springer, Heidelberg (2008) (invited talk)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Haralampos Tsaknakis
    • 1
  • Paul G. Spirakis
    • 1
    • 2
  1. 1.Research Academic Computer Technology Institute (RACTI)Greece
  2. 2.Dept. of Computer Eng. and InformaticsPatras UniversityPatrasGreece

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