Israel Journal of Mathematics

, Volume 92, Issue 1–3, pp 1–21 | Cite as

The existence of equilibria in certain games, separation for families of convex functions and a theorem of Borsuk-Ulam type

  • R. S. Simon
  • S. Spież
  • H. Toruńczyk


The existence of a Nash equilibrium in the undiscounted repeated two-person game of incomplete information on one side is established. The proof depends on a new topological result resembling in some respect the Borsuk-Ulam theorem.


Nash Equilibrium Incomplete Information Inverse Limit Repeated Game Separation Theorem 
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  1. [Au-Ha] R. Aumann and S. Hart,Bi-convexity and bi-martingales, Israel Journal of Mathematics54 (1986), 159–180.MATHCrossRefMathSciNetGoogle Scholar
  2. [Au-Mal] R. Aumann and M. Maschler,Game Theoretic Aspects of Gradual Disarmament, Mathematica V.1–V.55, Princeton N.J. (1966) (to be reprinted in [Au-Ma2]).Google Scholar
  3. [Au-Ma2] R. Aumann and M. Maschler,Repeated Games with Incomplete Information, prepared for M.I.T. Press.Google Scholar
  4. [Au-Ma-St] R. Aumann, M. Maschler and R. Stearns,Repeated games of incomplete information: An approach to the non-zero-sum case, Mathematica ST-143, Ch IV, 117–216, Princeton, N.J. (1968) (to be reprinted in [Au-Ma2]).Google Scholar
  5. [Be] E. Begle,The Vietoris mapping theorem for bicompact spaces, Annals of Mathematics51 (1950), 534–543.CrossRefMathSciNetGoogle Scholar
  6. [Bl] D. Blackwell,An analog of the minimax theorem for vector payoffs, Pacific Journal of Mathematics6 (1956), 1–8.MATHMathSciNetGoogle Scholar
  7. [Bo] K. Borsuk,Drei Sätze über die n-dimensionale Euklidische Sphäre, Fundamenta Mathematicae20 (1933), 177–190.MATHGoogle Scholar
  8. [ES] S. Eilenberg and N. Steenrod,Foundations of Algebraic Topology, Princeton Univ. Press, 1952.Google Scholar
  9. [Fo] F. Forges,Repeated games of incomplete information: non-zero-sum, inHandbook of Game Theory, Vol. 1 (R. J. Aumann and S. Hart, eds.), Elsevier Science Publishers, Amsterdam, 1992.Google Scholar
  10. [Hars] J. Harsanyi,Games with incomplete information played by Bayesian players, parts I–III, Management Science14 (1967/68) 159–182, 320–334, 486–502.MathSciNetMATHGoogle Scholar
  11. [Hart] S. Hart,Non-zero-sum two-person repeated games with incomplete information, Mathematics of Operations Research10 (1985), 117–153.MATHMathSciNetCrossRefGoogle Scholar
  12. [Hav] W. E. Haver,A near-selection theorem, General Topology and its Applications9 (1978), 117–124.CrossRefMathSciNetGoogle Scholar
  13. [Jo] K. D. Joshi,A non-symmetric generalization of the Borsuk-Ulam theorem, Fundamenta Mathematicae80 (1973), 13–33.MATHMathSciNetGoogle Scholar
  14. [Ku] H. W. Kuhn,Extensive Games and the Problem of Information, inContribution to the Theory of Games II, Annals of Mathematics Studies28 (1953), 193–216.MathSciNetGoogle Scholar
  15. [Ol] J. Olędzki,On a generalization of the Borsuk-Ulam l’Académie Polonaise des Sciences 26 (1978), 157–162.MATHGoogle Scholar
  16. [Ro] R. T. Rockafeller,Convex Analysis, Princeton Univ. Press, 1970.Google Scholar
  17. [R-S] C. P. Rourke and B.J. Sanderson,Introduction to Piecewise-Linear Topology, Springer-Verlag, Berlin, 1972.MATHGoogle Scholar
  18. [Se] R. Selten,Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory4 (1974), 25–55.CrossRefMathSciNetGoogle Scholar
  19. [Si] K. Sieklucki,A generalization of the Borsuk-Ulam theorem on antipodal points, Bulletin de, l’Academie Polonaise des Sciences17 (1969), 629–631.MATHMathSciNetGoogle Scholar
  20. [So] S. Sorin,Some results on the existence of Nash equilibrium for non-zerosum games with incomplete information, International Journal of Game Theory12 (1983), 193–205.MATHCrossRefMathSciNetGoogle Scholar
  21. [Sp] E. H. Spanier,Algebraic Topology, McGraw-Hill, New York, 1966.MATHGoogle Scholar

Copyright information

© Hebrew University 1995

Authors and Affiliations

  1. 1.Institut für Mathematishe WirtschaftforschungUniversität BielefeldBielefeldGermany
  2. 2.Instytut Matematyczny PANWarszawaPoland

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