Abstract
The investigation of equilibrium existence in a salient class of multistage games is simplified and generalized by reducing them to singlestage games which correspond also to competitive economies. Thus, economic theory provides much of the motivation as well as method for the study.
Working in locally convex topological vector spaces, a fixedpoint theorem of Fan is applied to show the existence of the Nash equilibria studied. En route, but also as a matter of interest in itself, certain topological foundations of the equilibrium analysis used are laid out.
A particular feature of generality of the dynamic games studied is that the feasible control regions of individual players are allowed to depend on the past state-history and control-history. Another such feature is that the next-state map is allowed certain nonlinearities.
Similar content being viewed by others
References
Arrow, K. J., andDebreu, G.,Existence of an Equilibrium for a Competitive Economy, Econometrica, Vol. 22, No. 3, 1954.
Debreu, G.,A Social Equilibrium Existence Theorem, Proceedings of the National Academy of Sciences, Vol. 38, No. 10, 1952.
Eilenberg, S., andMontgomery, D.,Fixed Point Theorems for Multi-Valued Transformations, American Journal of Mathematics, Vol. 68, No. 2, 1946.
Sertel, M. R.,Elements of Equilibrium Methods for Social Analysis, Massachusetts Institute of Technology, Ph.D. Thesis, 1971.
Kuhn, A., andSzegö, G., Editors,The Theory of Differential Games and Related Topics, North Holland Publishing Company, Amsterdam, Holland, 1971.
Ho, Y. C.,Differential Games, Dynamic Optimization, and Generalized Control Theory, Journal of Optimization Theory and Applications, Vol. 6, No. 3, 1970.
Parthasarathy, K. P.,Probability Measures on Metric Spaces, Academic Press, New York, New York, 1967.
Michael, E.,Topologies on Spaces of Subsets, Transactions of the American Mathematical Society, Vol. 71, No. 4, 1951.
Fan, K.,Fixed-Point and Minimax Theorems in Locally Convex Topological Linear Spaces, Proceedings of the National Academy of Sciences, Vol. 38, No. 2, 1952.
Hogan, W. F.,Point to Set Maps in Mathematical Programming, SIAM Review, Vol. 5, No. 3, 1973.
Kleindorfer, P., andSertel, M. R.,Equilibrium Existence Results for Simple Economies and Dynamic Games, International Institute of Management, West Berlin, Germany, Preprint No. I/72-22, 1972.
Starr, A. W., andHo, Y. C.,Nonzero-Sum Differential Games, Journal of Optimization Theory and Applications, Vol. 3, No. 3, 1969.
Starr, A. W., andHo, Y. C.,Further Properties of Nonzero-Sum Differential Games, Journal of Optimization Theory and Applications, Vol. 4, No. 4, 1969.
Prakash, P., andSertel, M. R.,Fixed Point and Minimax Theorems in Semi-Linear Spaces, Massachusetts Institute of Technology, Sloan School of Management, Working Paper, No. 484-70, 1970.
Author information
Authors and Affiliations
Additional information
Communicated by P. P. Varaiya
This paper is based on the more complete paper (Ref. 11) which was presented at the IEEE Conference on Decision and Control in New Orleans, 1972. The authors believe that the more complete version (Ref. 11) also makes the connection between dynamic games and economies rather more transparent than does the present version and that, from the mathematical economic theorist's viewpoint, Ref. 11 is probably to be considered as a more proper exposition.
Rights and permissions
About this article
Cite this article
Kleindorfer, P.R., Sertel, M.R. Equilibrium existence results for simple dynamic games. J Optim Theory Appl 14, 613–631 (1974). https://doi.org/10.1007/BF00932964
Issue Date:
DOI: https://doi.org/10.1007/BF00932964