Advertisement

On Extensions of the Cournot-Nash Theorem

Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 244)

Abstract

This paper is devoted to a problem that occupies a central position in economic theory and whose origins lie in Augustin Cournot’s Recherches sur les Principles Mathématiques de la Théorie des Richesses. The setting is that of a group of players each of which pursues his self-interest as reflected in an individual pay-off function defined on an individual strategy set. What makes the problem interesting is that the optimum choice of any player depends on the actions of all the other players, this dependence being reflected in the pay-off function or in the strategy set or both. As such, this is a problem par excellence in what is now termed non-cooperative game theory.

Keywords

Banach Space Nash Equilibrium Separable Banach Space Weak Compactness Pure Strategy Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K. J. Arrow and G. Debreu (1954), Existence of equilibrium for a competitive economy, Econometrica 22, 265–290.CrossRefGoogle Scholar
  2. Z. Artstein (1979), A note on Fatou’s Lemma in several dimensions, J. Math. Economics 6, 277–282.CrossRefGoogle Scholar
  3. R. J. Aumann (1964), Markets with a continuum of traders, Econometrica 32, 39–50.CrossRefGoogle Scholar
  4. R. J. Aumann (1965), Integrals of set-valued functions, J. of Math. Analysis and Applications 12, 1–12.CrossRefGoogle Scholar
  5. R. J. Aumann (1966), Existence of competitive equilibria in markets with a continuum of agents, Econometrica 34, 1–17.CrossRefGoogle Scholar
  6. C. Berge (1963), Topological Spaces, Oliver and Boyd, Edinburg.Google Scholar
  7. T. F. Bewley (1973), The equality of the core and the set of equilibria with infinitely many commodities and a continuum of agents, International Economic Review 14, 383–396.CrossRefGoogle Scholar
  8. F. E. Browder (1968), The fixed point theory of multi-valued mappings in topological vector spaces, Math. Annalen 177, 283–301.CrossRefGoogle Scholar
  9. C. Castaing and M. Valadier (1977), Convex Analysis and Measurable Multi-functions, Lecture Notes in Mathematics No. 480, Springer-Verlag, New York.Google Scholar
  10. A. A. Cournot (1838), Recherches sur les Principles Mathématiques de la Théorie des Richesses, librairie des sciences politiques et sociales, Paris. Also translation by Nathaniel Bacon (1897), Macmillan, New York.Google Scholar
  11. G. Debreu (1952), A social equilibrium existence theorem, Proceedings of the National Academy of Sciences of the U.S.A. 38, 886–893.CrossRefGoogle Scholar
  12. J. Diestel (1977), Remarks on weak compactness in L1(μ,X), Glasgow Math. Journal 18, 87–91.CrossRefGoogle Scholar
  13. J. Diestel (1984), Sequences and Series in Banach spaces, Springer-Verlag, New York.CrossRefGoogle Scholar
  14. J. Diestel and J. J. Uhl, Jr. (1977), Vector Measures, Mathematical Surveys No. 15, American Mathematical Society, Rhode Island.Google Scholar
  15. J. Dieudonné (1970), Treatise on Analysis, Vol. II, Academic Press, San Francisco.Google Scholar
  16. N. Dinculeanu (1973), Linear operations on Lp-spaces, in D.H. Tucker and H. B. Maynard (Eds.), Vector and Operator Valued Measures and Applications, Academic Press, New York,Google Scholar
  17. P. Dubey and L. Shapley (1977), Noncooperative exchange with a continuum of traders, Cowles Foundation Discussion Paper No. 447. Google Scholar
  18. N. Dunford and J. T. Schwartz (1958), Linear Operators: Part I., John Wiley Publishing Co., New York.Google Scholar
  19. Ky Fan (1952), Fixed points and minimax theorems in locally convex linear spaces, Proceedings of the National Academy of Sciences of the U.S.A. 38, 121–126.CrossRefGoogle Scholar
  20. Ky Fan (1966), Applications of a theorem concerning sets with convex sections, Math. Annalen 163, 189–203.CrossRefGoogle Scholar
  21. I. L. Glicksberg (1952), A further generalization of the Kakutani fixed point theorem with applications to Nash equilibrium points, Proceedings of the American Mathematical Society 3, 170–174.Google Scholar
  22. S. Hart, W. Hildenbrand and E. Kohlberg (1974), On equilibrium allocations as distributions on the commodity space, J. Math. Economics 1, 159–166.CrossRefGoogle Scholar
  23. W. Hildenbrand (1974), Core and Equilibria of a Large Economy, Princeton University Press, Princeton.Google Scholar
  24. S. Kakutani (1941), A generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal 8, 457–458.CrossRefGoogle Scholar
  25. J. L. Kelley and I. Namioka (1963), Linear Topological Spaces, Springer-Verlag Gratuate Text in Mathematics No. 36, Springer-Verlag, New York.Google Scholar
  26. M. Ali Khan (1982a), On the integration of set-valued mappings in a non-reflexive Banach space, John Hopkins Working Paper No. 98. Google Scholar
  27. M. Ali Khan (1982b), Equilibrium points of nonatomic games over a non-reflexive Banach space, John Hopkins Working Paper No. 100, forthcoming in the Journal of Approximation Theory, Google Scholar
  28. M. Ali Khan (1983), Equilibrium points of nonatomic games over a Banach space, presented at the N.S.F.-N.B.E.R. Conference on Mathematical Economics in April 1983.Google Scholar
  29. M. Ali Khan (1984), An alternative proof of Diestel’s theorem, Glasgow Math. Journal 25, 45–46.CrossRefGoogle Scholar
  30. M. Ali Khan and R. Vohra (1984), Equilibrium in abstract economies without ordered preferences and with a measure of agents, J. Math. Economics 13, 133–142.CrossRefGoogle Scholar
  31. M. Ali Khan and M. Majumdar (1984), Weak sequential convergence in L1(μ,X) and an approximate version of Fatou’s Lemma, Cornell University preprint. (Forthcoming in J. Math. Analysis and Applications.) Google Scholar
  32. T. Ma (1969), On sets with convex sections, J. Math. Analysis and Applications 27, 413–416.CrossRefGoogle Scholar
  33. A. Mas-Colell (1974), An equilibrium existence theorem without complete or transitive preferences, J. Math. Economics 1, 237–246.CrossRefGoogle Scholar
  34. A. Mas-Colell (1975), A model of equilibrium with differentiated commodities, J. Math. Economics 2, 263–295.CrossRefGoogle Scholar
  35. A. Mas-Colell (1983), On a theorem of Schmeidler, Harvard University preprint.Google Scholar
  36. J. F. Nash (1950), Equilibrium points in N-person games, Proceedings of the National Academy of Sciences of the U.S.A. 36, 48–49.CrossRefGoogle Scholar
  37. J. F. Nash (1951), Non-cooperative games, Annals of Mathematics 54, 286–295.CrossRefGoogle Scholar
  38. H. P. Rosenthal (1970), On injective Banach spaces and the spaces L(μ) for finite measures μ, Acta Mathematika 124, 205–248.CrossRefGoogle Scholar
  39. M. F. Saint-Beuve (1974), On the extension of von-Neumann-Aumann’s theorem, J. of Functional Analysis 17, 112–129.CrossRefGoogle Scholar
  40. D. Schmeidler (1973), Equilibrium points of nonatomic games, J. of Statistical Physics 7, 295–300.CrossRefGoogle Scholar
  41. L. Schwartz (1973), Randon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Oxford University Press, Bombay.Google Scholar
  42. W. Shafer (1974), The nontransitive consumer, Econometrica 42, 355–381.CrossRefGoogle Scholar
  43. W. Shafer and H. Sonnenschein (1975), Equilibrium in abstract economies without ordered preferences, J. Math. Economics.2, 345–348.CrossRefGoogle Scholar
  44. S. Toussaint (1982), On the existence of equilibrium with infinitely many commodities, Mannheim Working paper. Also forthcoming in Journal of Economic Theory. Google Scholar
  45. A. Ionescu Tulcea and C. Ionescu Tulcea (1962), On the lifting property II, Journal of Mathematics and Mechanics 11, 773–795.Google Scholar
  46. A. Wilansky (1978), Modern Methods in Topological Vector Spaces, McGraw-Hill Publishing Co., New York.Google Scholar
  47. N. Yannelis and N. Prabhaker (1983a), Existence of maximal elements and equilibria in linear topological spaces, J. Math. Economics 12, 233–245.CrossRefGoogle Scholar
  48. N. Yannelis and N. Prabhaker (1983b), Equilibrium in abstract economies with an infinite number of agents, an infinite number of commodities and without ordered preferences, Wayne State University preprint.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of IllinoisChampaignUSA

Personalised recommendations