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Nash’s “One-Page Proof”

Kakutani’s Set-Valued Fixed-Point Theorem
  • Joel H. Shapiro
Chapter
Part of the Universitext book series (UTX)

Overview

In this chapter we’ll study Shizuo Kakutani’s extension of the Brouwer Fixed-Point Theorem to maps whose values are sets, and we’ll show how John Nash used Kakutani’s result to provide a very quick proof of his famous Theorem  5.11 on the existence of Nash Equilibrium.

Keywords

Nash Equilibrium Probability Vector Continuous Selection Strategy Pair Finite Game 
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.

References

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    Bressan, A.: Noncooperative Differential Games. A Tutorial. Lecture Notes available online at http://www.math.psu.edu/bressan/PSPDF/game-lnew.pdf (2010)
  2. 23.
    Cellina, A.: Approximation of set valued functions and fixed point theorems. Ann. Mat. Pura Appl. 82(4), 17–24 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
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    Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8, 457–459 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 85.
    Nash Jr., J.F.: Equilibrium points in N-person games. Proc. Natl. Acad. Sci. U.S.A. 36, 48–49 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 89.
    von Neumann, J.L.: A model of general economic equilibrium. Rev. Econ. Studies 13, 1–9 (1945)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Joel H. Shapiro
    • 1
  1. 1.Portland State UniversityPortlandUSA

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