The Existence and Uniqueness of Equilibria in Convex Games with Strategies in Hilbert Spaces

  • Dean A. Carlson
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 6)

Abstract

In this paper we extend the approach of Rosen [7] for existence and uniqueness of Nash equilibria for finite-dimensional convex games to an abstract setting in which the strategies of each player are in separable Hilbert spaces. Through the use of an extension of the Kakutani fixed-point theorem, we are able to extend Rosen’s existence result to this setting. Our uniqueness results are obtained by extending Rosen’s notion of strict diagonal convexity to this setting. Several examples, in the context of open-loop dynamic games, to which our results may be applied are presented.

Keywords

Assure Nash Rosen Librium Rium 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Aubin, J-P. Applied functional analysis, Pure and Applied Mathematics, John Wiley and Sons, New York, 1979.Google Scholar
  2. [2]
    Bohnenblust, H. F. and Karlin, S. On a theorem of ville, in Contributions to the Theory of Games, Vol. 1, H. W. Kuhn and A. W. Tucker, eds., Princeton University Press, Princeton, pp. 155–160, 1950.Google Scholar
  3. [3]
    Carlson, D. A. and Haurie, A. A turnpike theory for infinite horizon open-loop differential games with decoupled controls, in New Trends in Dynamic Games and Applications, G. J. Olsder, ed., Annals of the International Society of Dynamic Games, Birkhäuser, Boston, pp. 353–376, 1995.Google Scholar
  4. [4]
    Carlson, D. A. and Haurie, A. A turnpike theory for infinite horizon open-loop competitive processes, SIAM J. Cont. Opt. 34 (4):1405–1419, 1996.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Cesari, L. Optimization-theory and applications: Problems with ordinary differential equations, Applications of Applied Mathematics, Vol. 17, Springer-Verlag, New York, 1983.Google Scholar
  6. [6]
    Haurie, A., Moresino, F., and Pourtalier, O. Oligopolies as Dynamic Games: A computational economics perspective, in P. Krall and H.-J. Luethi, eds., Operations Research—Ongoing Progress Pro. OR 98, Int. Conf Oper. Res.,ETH Zurich, Springer-Verlag, 31 August-3 September 1998.Google Scholar
  7. [7]
    Rosen, J.B. Existence and uniqueness of equilibrium points for concave n-person games, Econometrica 33 (3):520–534, 1965.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Dean A. Carlson
    • 1
  1. 1.Department of MathematicsUniversity of ToledoToledoUSA

Personalised recommendations