Abstract
The present note extends Debreu's equilibrium existence theorem for a generalized game in the context of finite-dimensional strategy spaces, by weakening the upper Semicontinuity and closed-valuedness assumption on the feasible strategy multifunctions. This is made by establishing an inequality of Ky Fan's type, whose proof is based on a selection theorem by E. Michael. An extension to generalized games with unbounded strategy spaces is also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aubin JP (1993) Optima and equilibria. Springer-Verlag, Berlin
Aubin JP, Frankowska H (1990) Set-valued analysis. Birkhäuser Boston
Cubiotti P (1992) Finite-dimensional quasi-variational inequalities associated with discontinuous functions. J Optim Theory Appl 72: 577–582
Cubiotti P (1993) An existence theorem for generalized quasi-variational inequalities. Set-Valued Anal 1:81–87
Debreu G (1952) A social equilibrium existence theorem. Proc Nat Acad Sci USA 38: 386–393
Glicksberg IL (1952) A further generalization of the Kakutany fixed point theorem with applications to Nash equilibrium points. Proc Amer Math Soc 38: 170–174
Harker PT (1991) Generalized Nash games and quasi-variational inequalities. European J Oper Res 54: 81–94
Harker PT, Pang JS (1990) Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms, and applications. Math Programming 48: 161–220
Ichiishi T (1983) Game theory for economic analysis. Academic Press, New York
Klein E, Thompson AC (1984) Theory of correspondences. John Wiley and Sons, New York
Michael E (1956) Continuous selections I. Ann of Math 63: 361–382
Naselli Ricceri O (1991) On the covering dimension of the fixed point set of certain multifunctions. Comment Math Univ Carolin 32: 281–286
Nash J (1950) Equlibrium points in N-person games. Proc Nat Acad Sci USA 36: 48–49
Nash J (1951) Non-cooperative games. Ann of Math 54: 286–295
Nikaido H, Isoda K (1955) Note on noncooperative convex games. Pacific J Math 4: 65–72
Tan KK, Yuan XZ (1994) Lower semicontinuity of multivalued mappings and equilibrium points. Proceedings of the First World Congress of Nonlinear Analysis, Walter de Gruyter Publishers
Tian G, Zhou J (1992) The maximum theorem and the existence of Nash equilibrium of (generalized) games without lower semicontinuities. J Math Anal Appl 166: 351–364
Yannelis NC (1987) Equilibria in noncooperative models of competition. J Economic Theory 41: 96–111
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cubiotti, P. Existence of nash equilibria for generalized games without upper semicontinuity. Int J Game Theory 26, 267–273 (1997). https://doi.org/10.1007/BF01295855
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01295855