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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
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.
Debreu, G.: Economies with a Finite Number of Equilibria. Econometrica,38, 387–392, 1970.
Harsanyi, J. C.: Games with Randomly Disturbed Payoffs. International Journal of Game Theory,2, 1–23, 1973.
Nash, J. F.: Noncooperative Games. Annals of Mathematics,54, 286–295, 1951.
Sard, A.: A Measure of Critical Values of Differentiable Maps. Bulletin of the Mathematical Society,48, 883–890, 1942.
Van Der Waerden, B. L.: Einführung in die algebraische Geometrie. Berlin, 1939.
Wilson, R.: Computing Equilibria inN-person Games. SIAM Journal of Applied Mathematics,21, 80–87, 1971.
This research has been supported by Grant GS-3222 of the National Science Foundation, through the Center for Research in Management Science, University of California, Berkeley.
About this article
Cite this article
Harsanyi, J.C. Oddness of the number of equilibrium points: A new proof. Int J Game Theory 2, 235–250 (1973). https://doi.org/10.1007/BF01737572
- Equilibrium Point
- Economic Theory
- Game Theory
- Payoff Function
- Topological Property