International Journal of Game Theory

, Volume 2, Issue 1, pp 235–250 | Cite as

Oddness of the number of equilibrium points: A new proof

  • J. C. Harsanyi
Papers

Abstract

A new proof is offered for the theorem that, in “almost all” finite games, the number of equilibrium points isfinite andodd. The proof is based on constructing a one-parameter family of games with logarithmic payoff functions, and studying the topological properties of the graph of a certain algebraic function, related to the graph of the set of equilibrium points for the games belonging to this family. In the last section of the paper, it is shown that, in the space of all games of a given size, those “exceptional” games which fail to satisfy the theorem (by having an even number or an infinity of equilibrium points) is a closed set of measure zero.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aumann, R. J.: Acceptable Points in General Cooperativen-person Games. Contributions to the Theory of Games, IV (edited by A. W. Tucker and R. D. Luce). Princeton, N. J., pp. 287–324, 1959.Google Scholar
  2. Debreu, G.: Economies with a Finite Number of Equilibria. Econometrica,38, 387–392, 1970.Google Scholar
  3. Harsanyi, J. C.: Games with Randomly Disturbed Payoffs. International Journal of Game Theory,2, 1–23, 1973.Google Scholar
  4. Nash, J. F.: Noncooperative Games. Annals of Mathematics,54, 286–295, 1951.Google Scholar
  5. Sard, A.: A Measure of Critical Values of Differentiable Maps. Bulletin of the Mathematical Society,48, 883–890, 1942.Google Scholar
  6. Van Der Waerden, B. L.: Einführung in die algebraische Geometrie. Berlin, 1939.Google Scholar
  7. Wilson, R.: Computing Equilibria inN-person Games. SIAM Journal of Applied Mathematics,21, 80–87, 1971.Google Scholar

Copyright information

© Physica-Verlag Rudolf Liebing KG 1973

Authors and Affiliations

  • J. C. Harsanyi
    • 1
  1. 1.University of CaliforniaBerkeley

Personalised recommendations