Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Alternative version of Shapley's theorem on closed coverings of a simplex


Author: Tatsuro Ichiishi
Journal: Proc. Amer. Math. Soc. 104 (1988), 759-763
MSC: Primary 47H10; Secondary 54B99, 90D12
DOI: https://doi.org/10.1090/S0002-9939-1988-0964853-5
MathSciNet review: 964853
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Sperner's theorem as formulated by Ky Fan is dual to the KnasterKuratowski-Mazurkiewicz theorem. Shapley's theorem is a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem. This paper points out that Shapley's theorem is a generalization of Sperner's theorem as well, by establishing an alternative version of Shapley's theorem. Applications to the multiperson cooperative game theory are also discussed.


References [Enhancements On Off] (What's this?)

  • [1] L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1911), no. 1, 97–115 (German). MR 1511644, https://doi.org/10.1007/BF01456931
  • [2] Ky Fan, A covering property of simplexes, Math. Scand. 22 (1968), 17–20. MR 0240800, https://doi.org/10.7146/math.scand.a-10867
  • [3] Ky Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240. MR 0251603, https://doi.org/10.1007/BF01110225
  • [4] Ky Fan, A minimax inequality and applications, Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, 1972, pp. 103–113. MR 0341029
  • [5] Tatsuro Ichiishi, On the Knaster-Kuratowski-Mazurkiewicz-Shapley theorem, J. Math. Anal. Appl. 81 (1981), no. 2, 297–299. MR 622818, https://doi.org/10.1016/0022-247X(81)90063-9
  • [6] Tatsuro Ichiishi, Game theory for economic analysis, Economic Theory, Econometrics, and Mathematical Economics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. MR 700688
  • [7] Shizuo Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Math. J. 8 (1941), 457–459. MR 0004776
  • [8] H. Keiding and L. Thorlund-Petersen, The core of a cooperative game without side payments, mimeo, August 1985.
  • [9] B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für $ n$-dimensionale Simplexe, Fund. Math. 14 (1929), 132-137.
  • [10] C. E. Lemke and J. T. Howson Jr., Equilibrium points of bimatrix games, J. Soc. Indust. Appl. Math. 12 (1964), 413–423. MR 0173556
  • [11] Herbert E. Scarf, The core of an 𝑁 person game, Econometrica 35 (1967), 50–69. MR 0234735, https://doi.org/10.2307/1909383
  • [12] L. S. Shapley, On balanced games without side payments, Mathematical programming (Proc. Advanced Sem., Univ. Wisconsin, Madison, Wis., 1972) Academic Press, New York, 1973, pp. 261–290. Math. Res. Center Publ., No. 30. MR 0389244
  • [13] -, Lecture Notes, Dept. of Math., Univ. of California, Los Angeles, 1987.
  • [14] E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg 6 (1928), 265-272.
  • [15] M. Todd, Lecture Notes, School of Operations Research and Industrial Engineering, Cornell Univ., Ithaca, 1978.
  • [16] -, Private communication, 1979.
  • [17] R. Vohra, On Scarf's theorem on the non-emptiness of the core: A direct proof through Kakutani's fixed-point theorem, Working Paper No. 87-2, Dept. of Economics, Brown Univ., January 1987. Revised: June 1987.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H10, 54B99, 90D12

Retrieve articles in all journals with MSC: 47H10, 54B99, 90D12


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0964853-5
Article copyright: © Copyright 1988 American Mathematical Society