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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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 Backing Up

 Backing Up

Chapter:
(p.39) 2 Backing Up
Source:
Playing for Real
Author(s):

Ken Binmore (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780195300574.003.0002

This chapter's title signifies not only that backward induction is introduced, but that some backing up is necessary to put the material of the previous chapter on a more solid foundation. The game of Matching Pennies, with and without peeking, is used to illustrate the difference between games of perfect and imperfect competition. The rules of a game of a general game of perfect information are then described using the idea of a game tree. The principle of backward induction is introduced for the case of win-or-lose games, where its application is entirely uncontroversial. This leads to the notion of the value of a win-or-lose game. As examples, the values of the games of Nim and Hex are found. Chess is then introduced as an example of a strictly competitive game with more than two outcomes. It is shown that finite strictly competitive games always have a value. The connexion between Nash equilibria and saddle points is explained. Subgame-perfect equilibria are introduced as the end-product of backward induction. It is explained why some Nash equilibria are not subgame-perfect. The chapter ends with a discussion of the rationality assumption, and why subgame-perfect play may not always be a good idea when an opponent fails to behave rationally.

Keywords:   win-or-lose games, perfect information, game tree, backward induction, subgame, value of a game, Hex, Nim, Nash equilibrium, saddle point

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