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Playing for RealGame Theory$
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Ken Binmore

Print publication date: 2007

Print ISBN-13: 9780195300574

Published to Oxford Scholarship Online: May 2007

DOI: 10.1093/acprof:oso/9780195300574.001.0001

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 Keeping Your Balance

 Keeping Your Balance

Chapter:
(p.253) 8 Keeping Your Balance
Source:
Playing for Real
Author(s):

Ken Binmore (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780195300574.003.0008

This chapter focuses on the existence of Nash equilibria. The existence problem is first introduced with pure-strategy reaction curves for Noisy and Silent Duel. A proof of John Nash's existence theorem is then sketched using the Kakutani fixed-point theorem. The latter may be deduced from Brouwer's more famous fixed-point theorem. A sketch of a proof of Brouwer's theorem is offered using the fact that Hex cannot end in a draw. Games often have many Nash equilibria, which creates an equilibrium selection problem. Various possible approaches to the equilibrium selection problem are reviewed, including the evolutionary approach and Thomas Schelling's notion of a focal point.

Keywords:   Nash equilibrium, Silent Duel, Noisy Duel, Kakutani's fixed-point theorem, Nash's theorem, Brouwer's fixed-point theorem, Hex, equilibrium selection problem, David Hume, group selection

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