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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

(p.39) 2 Backing Up
Playing for Real

Ken Binmore (Contributor Webpage)

Oxford University Press

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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