The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Fixed Point Theorems

  • Andrew McLennan
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_646

Abstract

This article gives statements of the Tarski fixed point theorem and the main versions of the topological fixed point principle that have been applied in economic theory. Pointers are given to literature concerned with proofs of Brouwer’s theorem, and with algorithms for computing approximate fixed points. The topological results are all consequences of a slightly weakened version of the Eilenberg and Montgomery (American Journal of Mathematics 68: 214–222, 1946) fixed point theorem. The axiomatic characterization of the Leray–Schauder fixed point index (which is even more powerful) is also stated, and its application to issues concerning robustness of sets of equilibria is explained.

Keywords

Absolute neighbourhood retract Algebraic topology Brouwer’s fixed point th Contraction mapping th Convexity Cooperative game theory Cooperative game theory (core) Debreu–Gale–Kuhn–Nikaido lemma Eilenberg–Montgomery th Essential sets of fixed points Excess demand Existence of equilibrium Fixed point property Fixed point theorems Homotopy methods Hopf’s th Kakutani’s th Kinoshita’s th K–K–M–S th Lefschetz fixed point th Leray–Schauder fixed point index Nash equilibrium Perfect equilibrium Sard’s th Scarf algorithm Schauder fixed point th Sperner’s lemma Strategic stability Tarski’s fixed point th 
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Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Andrew McLennan
    • 1
  1. 1.