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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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 Fighting It Out

 Fighting It Out

(p.215) 7 Fighting It Out
Playing for Real

Ken Binmore (Contributor Webpage)

Oxford University Press

This chapter describes the theory of two-person, zero-sum games invented by John Von Neumann in 1928. It begins with an application to the computation of economic shadow prices. It shows that a two-person game is strictly competitive if, and only if, it has a zero-sum representation. Such a game can be represented using only the first player's payoff matrix. The minimax and maximin values of the matrix are defined and linked to the concept of a saddle point. The ideas are then related to a player's security level in a game. An inductive proof of Von Neumann's minimax theorem is offered. The connexion between the minimax theorem and the duality theorem of linear programming is explained. The method of solving certain two-person, zero-sum games geometrically with the help of the theorem of the separating hyperplane is introduced. The Hide-and-Seek Game is used as a non-trivial example.

Keywords:   strictly competitive game, shadow price, zero-sum game, matrix game, saddle point, security level, John Von Neumann, minimax theorem, duality theorem, separating hyperplane

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