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Noncooperative Game TheoryAn Introduction for Engineers and Computer Scientists$
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João P. Hespanha

Print publication date: 2017

Print ISBN-13: 9780691175218

Published to Princeton Scholarship Online: May 2018

DOI: 10.23943/princeton/9780691175218.001.0001

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Minimax Theorem

Minimax Theorem

Chapter:
(p.52) Lecture 5 Minimax Theorem
Source:
Noncooperative Game Theory
Author(s):

João P. Hespanha

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691175218.003.0005

This chapter deals with the Minimax Theorem and its proof, which is based on elementary results from convex analysis. The theorem states that for every matrix A, the average security levels of both players coincide. In a mixed policy, the min and max always commute. For every constant c, at least one of the players can guarantee a security level of c. The chapter first considers the statement of the Minimax Theorem before discussing the convex hull and the Separating Hyperplane Theorem, one of the key results in convex analysis. It then demonstrates how to prove the Minimax Theorem and presents the proof. It also shows the consequences of the Minimax Theorem and concludes with a practice exercise related to symmetric games and the corresponding solution.

Keywords:   convex hull, Minimax Theorem, convex analysis, average security level, mixed policy, Separating Hyperplane Theorem, symmetry game, hyperplane

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