Table of Contents
ISRN Computational Mathematics
Volume 2012 (2012), Article ID 459459, 8 pages
Research Article

Constructive Proof of the Existence of Nash Equilibrium in a Finite Strategic Game with Sequentially Locally Nonconstant Payoff Functions

Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan

Received 5 August 2011; Accepted 19 September 2011

Academic Editor: T. Karakasidis

Copyright © 2012 Yasuhito Tanaka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. B. Kellogg, T. Y. Li, and J. Yorke, “A constructive proof of Brouwer fixed-point theorem and computational results,” SIAM Journal on Numerical Analysis, vol. 13, no. 4, pp. 473–483, 1976. View at Google Scholar · View at Scopus
  2. D. Bridges and F. Richman, Varieties of Constructive Mathematics, vol. 97 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1987.
  3. D. van Dalen, “Brouwer’s ε-fixed point from Sperner’s lemma,” Theoretical Computer Science, vol. 412, no. 28, pp. 3140–3144, 2011. View at Google Scholar
  4. V. P. Orevkov, “A constructive mapping of a square onto itself displacing every constructive point,” Soviet Mathematics, vol. 4, pp. 1253–1256, 1963. View at Google Scholar
  5. J. L. Hirst, “Notes on reverse mathematics and Brouwer’s fixed point theorem,” 2000,
  6. W. Veldman, “Brouwer's approximate fixed-point theorem is equivalent to Brouwer's fan theorem,” in Logicism, Intuitionism, and Formalism, S. Lindstrom, E. Palmgren, K. Segerberg, and V. Stoltenberg-Hansen, Eds., vol. 341, pp. 277–299, Springer, Dordrecht, Germany, 2009. View at Publisher · View at Google Scholar
  7. Y. Tanaka, “Constructive proof of Brouwer’s fixed point theorem for sequentially locally nonconstant functions,” In press,
  8. E. Bishop and D. Bridges, Constructive Analysis, vol. 279, Springer, Berlin, Germany, 1985.
  9. D. Bridges and L. Vîţã, Techniques of Constructive Mathematics, Springer, Berlin, Germany, 2006.
  10. F. E. Su, “Rental harmony: Sperner's lemma in fair division,” The American Mathematical Monthly, vol. 106, no. 10, pp. 930–942, 1999. View at Publisher · View at Google Scholar
  11. Y. Tanaka, “Equivalence between the existence of an approximate equilibrium in a competitive economy and Sperner’s lemma: a constructive analysis,” ISRN Applied Mathematics, vol. 2011, Article ID 384625, 15 pages, 2011. View at Publisher · View at Google Scholar
  12. J. Berger, D. Bridges, and P. Schuster, “The fan theorem and unique existence of maxima,” Journal of Symbolic Logic, vol. 71, no. 2, pp. 713–720, 2006. View at Google Scholar
  13. J. Nash, “Non-cooperative games,” Annals of Mathematics. Second Series, vol. 54, pp. 286–295, 1951. View at Google Scholar
  14. S. Takahashi, “The number of pure Nash equilibria in a random game with nondecreasing best responses,” Games and Economic Behavior, vol. 63, no. 1, pp. 328–340, 2008. View at Publisher · View at Google Scholar
  15. P. G. Spirakis, “A note on proofs of existence of Nash equilibria in finite strategic games, of two players,” Computer Science Review, vol. 3, no. 2, pp. 101–103, 2009. View at Publisher · View at Google Scholar · View at Scopus
  16. N. G. Pavlidis, K. E. Parsopoulos, and M. N. Vrahatis, “Computing Nash equilibria through computational intelligence methods,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 113–136, 2005. View at Publisher · View at Google Scholar
  17. W. Stanford, “On the number of pure strategy Nash equilibria in finite common payoffs games,” Economics Letters, vol. 62, no. 1, pp. 29–34, 1999. View at Publisher · View at Google Scholar
  18. V. Conitzer and T. Sandholm, “New complexity results about Nash equilibria,” Games and Economic Behavior, vol. 63, no. 2, pp. 621–641, 2008. View at Publisher · View at Google Scholar