Nash’s “One-Page Proof”

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


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.


Nash Equilibrium Probability Vector Continuous Selection Strategy Pair Finite Game 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

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